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Cornell University Library 
Q 175.P75S4 1914 

Science and method. 

924 012 248 179 

Cornell University 

The original of this book is in 
the Cornell University Library. 

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[IJThe Relativity of Space 

II. Mathematical Definitions and Education 
III. Mathematics and Logic 
IV". The New Logics 

V. The Last Efforts of the Logisticians 


Introduction ........ 9 

I. The Selection of Facts . . . .15 
II. The Future of Mathematics . . .25 

III. Mathematical Discovery .... 46 

IV. Chance 64 





I. Mechanics and Radium 199 

II. Mechanics and Optics 213 

IIL The New Mechanics and Astronomy . -235 


I. The Milky Way and the Theory of Gases . 253 

II. French Geodesy 270 

General Conclusions 284 


Henri Poincar^ was, by general agreement, the 
most eminent scientific man of his generation — more 
eminent, one is tempted to think, than any man of 
science now living. From the mere variety of the 
subjects which he illuminated, there is certainly no 
one who can appreciate critically the whole of his 
work. Some conception of his amazing comprehen- 
siveness may be derived from the obituary number of 
the Revue de Mitaphysique et de Morale (September 
1913), where, in the course of 130 pages, four eminent 
men — a philosopher, a mathematician, an astronomer, 
and a physicist — tell in outline the contributions which 
he made to their several subjects. In all we find the 
same characteristics — swiftness, comprehensiveness, 
unexampled lucidity, and the perception of recondite 
but fertile analogies. 

Poincare's philosophical writings, of which the pres- 
ent volume is a good example, are not those of a 
professional philosopher : they are the untrammelled 
reflections of a broad and cultivated mind upon the 
procedure and the postulates of scientific discovery. 
The writing of professional philosophers on such sub- 
jects has too often the deadness of merely external 
description ; Poincar6's writing, on the contrary, as 
the reader of this book may see in his account of 
mathematical invention, has the freshness of actual 
experience, of vivid, intimate contact with what he is 


describing. There results a certain richness and 
resonance in his words : the sound emitted is not 
hollow, but comes from a great mass of which only 
the polished surface appears. His wit, his easy mas- 
tery, and his artistic love of concealing the labour of 
thought, may hide from the non-mathematical reader 
the background of solid knowledge from which his 
apparent paradoxes emerge : often, behind what may 
seem a light remark, there lies a whole region of 
mathematics which he himself has helped to explore. 

A philosophy of science is growing increasingly 
necessary at the present time, for a variety of reasons. 
Owing to increasing specialization, and to the con- 
stantly accelerated accumulation of new facts, the 
general bearings of scientific systems become more 
and more lost to view, and the synthesis that depends 
on coexistence of multifarious knowledge in a single 
mind becomes increasingly difficult. In order to over- 
come this difficulty, it is necessary that, from time to 
time, a specialist capable of detachment from details 
should set forth the main lines and essential structure 
of his science as it exists at the moment. But it is 
not results, which are what mainly interests the man 
in the street, that are what is essential in a science : 
what is essential is its method, and it is with method 
that Poincar^'s philosophical writings are concerned. 

Another reason which makes a philosophy of science 
specially useful at the present time is the revolutionary 
progress, the sweeping away of what had seemed fixed 
landmarks, which has so far characterized this century, 
especially in physics. The conception of the " working 
hypothesis,'' provisional, approximate, and merely use- 
ful, has more and more pushed aside the comfortable 


eighteenth century conception of "laws of nature." 
Even the Newtonian dynamics, which for over two 
hundred years had seemed to embody a definite con- 
quest, must now be regarded as doubtful, and as 
probably only a first rough sketch of the ways of 
matter. And thus, in virtue of the very rapidity of 
our progress, a new theory of knowledge has to be 
sought, more tentative and more modest than that of 
more confident but less successful generations. Of 
this necessity Poincar6 was acutely conscious, and it 
gave to his writings a tone of doubt which was hailed 
with joy by sceptics and pragmatists. But he was in 
truth no sceptic : however conscious of the difficulty 
of attaining knowledge, he never admitted its impos- 
sibilit)'. " It is a mistake to believe," he said, " that the 
love of truth is indistinguishable from the love of cer- 
tainty ;" and again: "To doubt everything or to believe 
everything are two equally convenient solutions ; both 
dispense with the necessity of reflection." His was the 
active, eager doubt that inspires a new scrutiny, not 
the idle doubt that acquiesces contentedly in nescience- 
'Two opposite and conflicting qualities are required 
for the successful practice of philosophy — comprehen- 
siveness of outlook, and minute, patient analysis. Both 
exist in the highest degree in Descartes and Leibniz ; 
but in their day comprehensiveness was less difficult 
than it is now. Since Leibniz, I do not know of any 
philosopher who has possessed both : broadly speaking, 
British philosophers have excelled in analysis, while 
those of the Continent have excelled in breadth and 
scope. In this respect, Poincare is no exception : in 
philosophy, his mind was intuitive and synthetic; 
wonderfully skilful, it is true, in analysing a science 


until he had extracted its philosophical essence, and 
in combining this essence with those of other sciences, 
but not very apt in those further stages of analysis 
which fall within the domain of philosophy itself. He 
built wonderful edifices with the philosophic materials 
that he found ready to hand, but he lacked the patience 
and the minuteness of attention required for the crea- 
tion of new materials. For this reason, his philosophy, 
though brilliant, stimulating, and instructive, is not 
among those that revolutionize fundamentals, or com- 
pel us to remould our imaginative conception of the 
nature of things. In fundamentals, broadly speaking, 
he remained faithful to the authority of Kant. 

Readers of the following pages will not be surprised 
to learn that his criticisms of mathematical logic do 
not appear to me to be among the best parts of his 
work. He was already an old man when he became 
aware of the existence of this subject, and he was led, by 
certain indiscreet advocates, to suppose it in some way 
opposed to those quick flashes of insight in mathe- 
matical discovery which he has so admirably described- 
No such opposition in fact exists ; but the misconcep- 
tion, however regrettable, was in no way surprising. 

To be always right is not possible in philosophy ; 
but Poincar^'s opinions, right or wrong, are always the 
expression of a powerful and original mind with a 
quite unrivalled scientific equipment ; a masterly style, 
great wit, and a profound devotion to the advance- 
ment of knowledge. Through these merits, his books 
supply, better than any others known to me, the 
growing need for a generally intelligible account of 
the philosophic outcome of modern science. 

Bertrand Russell. 


In this work I have collected various studies which are 
more or less directly concerned with scientific metho- 
dology. [The scientific method consists in observation 
and experiment. If the scientist had an infinity of 
time at his disposal, it would be sufficient to say to 
him, " Look, and look carefully." But, since he has 
not time to look at everything, and above all to look 
carefully, and since it is better not to look at all than 
to look carelessly, he is forced to make a selection. 
The first question, then, is to know how to make this 
selection. This question confronts the physicist as 
well as the historian ; it also confronts the mathema- 
tician, and the principles which should guide them all 
are not very dissimilar. The scientist conforms to 
them instinctively, and by reflecting on these principles 
one can foresee the possible future of mathematics. 

We shall understand this still better if we observe 
the scientist at work ; and, to begin with, we must have 
some acquaintance with the psychological mechanism 
of discovery, more especially that of mathematical dis- 
covery. Observation of the mathematician's method 
of working is specially instructive for the psychologist. 

In all sciences depending on observation, we must 


reckon with errors due to imperfections of our senses 
and of our instruments. Happily we may admit that, 
under certain conditions, there is a partial compensa- 
tion of these errors, so that they disappear in averages. 
This compensation is due to chance. But what is 
chance? It is a notion which is difficult of justilica- 
tion, and even of definition ; and yet what I have just 
said with regard to errors of observation, shows that 
the scientist cannot get on without it. It is necessary, 
therefore, to give as accurate a definition as possible 
of this notion, at once so indispensable and so elusive. 

These are generalities which apply in the main to 
all sciences. For instance, there is no appreciable 
difference between the mechanism of mathematical 
discovery and the mechanism of discovery in general. 
Further on I approach questions more particularly 
concerned with certain special sciences, beginning with 
pure mathematics. 

In the chapters devoted to them, I am obliged to 
treat of somewhat more abstract subjects, and, to begin 
with, I have to speak of the notion of space. Every one 
knows that space is relative, or rather every one says 
so, but how many people think still as if they con- 
sidered it absolute. Nevertheless, a little reflection 
will show to what contradictions they are exposed. 

Questions concerning methods of instruction are of 
importance, firstly, on their own account, and secondly, 
because one cannot reflect on the best method of 
imbuing virgin brains with new notions without, at 
the same time, reflecting on the manner in which 
these notions have been acquired by our ancestors, 
and consequently on their true origin — that is in 
reality, on their true nature. Why is it that, in most 


cases, the definitions which satisfy scientists mean 
nothing at all to children? Why is it necessary to 
give them other definitions ? This is the question I 
have set myself in the chapter which follows, and its 
solution might, I think, suggest useful reflections to 
philosophers interested in the logic of sciences. 

On the other hand, there are many geometricians 
who believe that mathematics can be reduced to the 
rules of formal logic. Untold efforts have been made 
in this direction. To attain their object they have not 
hesitated, for instance, to reverse the historical order of 
the genesis of our conceptions, and have endeavoured 
to explain the finite by the infinite. I think I have suc- 
ceeded in showing, for all who approach the problem 
with an open mind, that there is in this a deceptive 
illusion. I trust the reader will understand the im- 
portance of the question, and will pardon the aridity 
of the pages I have been constrained to devote to it. 

The last chapters, relating to mechanics and astron- 
omy, will be found easier reading. 

Mechanics seem to be on the point of undergoing a 
complete revolution. The ideas which seemed most 
firmly established are being shattered by daring 
innovators. It would certainly be premature to 
decide in their favour from the start, solely because 
they are innovators ; but it is interesting to state 
their views, and this is what I have tried to do. As 
far as possible I have followed the historical order, 
for the new ideas would appear too surprising if we 
did not see the manner in which they had come into 

Astronomy offers us magnificent spectacles, and 
raises tremendous problems. We cannot dream of 


applying the experimental method to them directly ; 
our laboratories are too small. But analogy with the 
phenomena which these laboratories enable us to reach 
may nevertheless serve as a guide to the astronomer. 
The Milky Way, for instance, is an assemblage of suns 
whose motions appear at first sight capricious. But 
may not this assemblage be compared with that of 
the molecules of a gas whose properties we have 
learnt from the kinetic theory of gases ? Thus the 
method of the physicist may come to the aid of the 
astronomer by a side-track. 

Lastly, I have attempted to sketch in a few lines the 
history of the development of French geodesy. I have 
■shown at what cost, and by what persevering efforts 
and often dangers, geodesists have secured for us the 
few notions we possess about the shape of the earth. 
Is this really a question of method ? Yes, for this 
history certainly teaches us what precautions must 
surround any serious scientific operation, and what 
time and trouble are involved in the conquest of a 
single new decimal. 




Tolstoi explains somewhere in his writings why, in 
his opinion, " Science for Science's sake " is an absurd 
conception. We cannot know all the facts, since they 
are practically infinite in number. We must make a 
selection ; and that being so, can this selection be 
-governed by the mere caprice of our curiosity? Is 
it not better to be guided by utility, by our practical, 
and more especially our moral, necessities ? Have we 
not some better occupation than counting the number 
of lady-birds in existence on this planet ? 

It is clear that for him the word utility has not the 
meaning assigned to it by business men, and, after 
them, by the greater number of our contemporaries. 
He cares but little for the industrial applications of 
science, for the marvels of electricity or of auto- 
mobilism, which he regards rather as hindrances to 
moral progress. For him the useful is exclusively 
what is capable of making men better. 

It is hardly necessary for me to state that, for my 
part, I could not be satisfied with either of these 
ideals. I have no liking either for a greedy and 
narrow plutocracy, or for a virtuous unaspiring 
democracy, solely occupied in turning the other 


cheek, in which we should find good people devoid of 
curiosity, who, avoiding all excesses, would not die 
of any disease — save boredom. But it is all a matter 
of taste, and that is not the point I wish to discuss. 

None the less the question remains, and it claims 
our attention. If our selection is only determined by 
caprice or by immediate necessity, there can be no 
science for science's sake, and consequently no science. 
Is this true? There is no disputing the fact that a 
selection must be made : however great our activity, 
facts outstrip us, and we can never overtake them ; 
while the scientist is discovering one fact, millions 
and millions are produced in every cubic inch of his 
body. Trying to make science contain nature is like 
trying to make the part contain the whole. 

But scientists believe that there is a hierarchy 
of facts, and that a judicious selection can be made. 
They are right, for otherwise there would be no science, 
and science does exist. One has only to open one's 
eyes to see that the triumphs of industry, which have 
enriched so many practical men, would never have 
seen the light if only these practical men had existed, 
and if they had not been preceded by disinterested 
fools who died poor, who never thought of the useful, 
and yet had a guide that was not their own caprice. 

What these fools did, as Mach has said, was to save 
their successors the trouble of thinking. If they had 
worked solely in view of an immediate application, 
they would have left nothing behind them, and in face 
of a new requirement, all would have had to be done 
again. Now the majority of men do not like thinking, 
and this is perhaps a good thing, since instinct guides 
them, and very often better than reason would o-uide 
(1.777) '^ 


a pure intelligence, at least whenever they are pursuing 
an end that is immediate and always the same. But 
instinct is routine, and if it were not fertilized by 
thought, it would advance no further with man than 
with the bee or the ant. It is necessary, therefore, to 
think for those who do not like thinking, and as they 
are many, each one of our thoughts must be useful 
in as many circumstances as possible. For this 
reason, the more general a law is, the greater is its 

This shows us how our selection should be made. 
The most interesting facts are those which can be 
used several times, those which have a chance of 
recurring. We have been fortunate enough to be born 
in a world where there are such facts. Suppose that 
instead of eighty chemical elements we had eighty 
millions, and that they were not some common and 
others rare, but uniformly distributed. Then each 
time we picked up a new pebble there would be a 
strong probability that it was composed of some un- 
known substance. Nothing that we knew of other 
pebbles would tell us anything about it. Before each 
new object we should be like a new-born child ; like 
him we could but obey our caprices or our necessities. 
In such a world there would be no science, perhaps 
thought and even life would be impossible, since 
evolution could not have developed the instincts of 
self-preservation. Providentially it is not so ; but this 
blessing, like all those to which we are accustomed, is 
not appreciated at its true value. The biologist would 
be equall}'- embarrassed if there were only individuals 
and no species, and if heredity did not make children 
resemble their parents. 

(1,777J 2 


Which, then, are the facts that have a chance of 
recurring? In the first place, simple facts. It is 
evident that in a complex fact many circumstances 
are united by chance, and that only a still more 
improbable chance could ever so unite them again. 
But are there such things as simple facts ? and if there 
are, how are we to recognize them ? Who can tell 
that what we believe to be simple does not conceal 
an alarming complexity? All that we can say is 
that we must prefer facts which appear simple, to 
those in which our rude vision detects dissimilar 
elements. Then only two alternatives are possible ; 
either this simplicity is real, or else the elements 
are so intimately mingled that they do not admit of 
being distinguished. In the first case we have a 
chance of meeting the same simple fact again, either 
in all its purity, or itself entering as an element into 
some complex whole. In the second case the intimate 
mixture has similarly a greater chance of being re- 
produced than a heterogeneous assemblage. Chance 
can mingle, but it cannot unmingle, and a combination 
of various elements in a well-ordered edifice in which 
something can be distinguished, can only be made 
deliberately. There is, therefore, but little chance that 
an assemblage in which different things can be dis- 
tinguished should ever be reproduced. On the other 
hand, there is great probability that a mixture which 
appears homogeneous at first sight will be reproduced 
several times. Accordingly facts which appear simple, 
even if they are not so in reality, will be more easily 
brought about again by chance. 

It is this that justifies the method instinctively 
adopted by scientists, and what perhaps justifies it 


still better is that facts which occur frequently appear 
to us simple just because we are accustomed to 

But where is the simple fact ? Scientists have tried 
to find it in the two extremes, in the infinitely great 
and in the infinitely small. The astronomer has found 
it because the distances of the stars are immense, so 
great that each of them appears only as a point and 
qualitative differences disappear, and because a point 
is simpler than a body which has shape and qualities. 
The physicist, on the other hand, has sought the 
elementary phenomenon in an imaginary division of 
bodies into infinitely small atoms, because the con- 
ditions of the problem, which undergo slow and con- 
tinuous variations as we pass from one point of the 
body to another, may be regarded as constant within 
each of these little atoms. Similarly the biologist has 
been led instinctively to regard the cell as more interest- 
ing than the whole animal, and the event has proved 
him right, since cells belonging to the most diverse 
organisms have greater resemblances, for those who can 
recognize them, than the organisms themselves. The 
sociologist is in a more embarrassing position. The 
elements, which for him are men, are too dissimilar, too 
variable, too capricious, in a word, too complex them- 
selves. Furthermore, history does not repeat itself; 
how, then, is he to select the interesting fact, the fact 
which is repeated ? Method is precisely the selection 
of facts, and accordingly our first care must be to 
devise a method. Many have been devised because 
none holds the field undisputed. Nearly every socio- 
logical thesis proposes a new method, which, however, 
its author is very careful not to apply, so that sociology 


is the science with the greatest number of methods 
and the least results. 

It is with regular facts, therefore, that we ought to 
begin ; but as soon as the rule is well established, as 
soon as it is no longer in doubt, the facts which are in 
complete conformity with it lose their interest, since 
they can teach us nothing new. Then it is the excep- 
tion which becomes important. We cease to look for 
resemblances, and apply ourselves before all else to 
differences, and of these differences we select first 
those that are most accentuated, not only because 
they are the most striking, but because they will be 
the most instructive. This will be best explained by a 
simple example. Suppose we are seeking to determine 
a curve by observing some of the points on it. The 
practical man who looked only to immediate utility 
would merely observe the points he required for some 
special object ; these points would be badly distributed 
on the curve, they would be crowded together in cer- 
tain parts and scarce in others, so that it would be 
impossible to connect them by a continuous line, and 
they would be useless for any other application. The 
scientist would proceed in a different manner. Since 
he wishes to study the curve for itself, he will distribute 
the points to be observed regularly, and as soon as he 
knows some of them, he will join them by a regular 
line, and he will then have the complete curve. But 
how is he to accomplish this ? If he has determined 
one extreme point on the curve, he will not remain 
close to this extremity, but will move to the other end. 
After the two extremities, the central point is the most 
instructive, and so on. 

Thus when a rule has been established, we have first 


to look for the cases in which the rule stands the best 
chance of being found in fault. This is one of many 
reasons for the interest of astronomical facts and of 
geological ages. By making long excursions in 
space or in time, we may find our ordinary rules 
completely upset, and these great upsettings will give 
us a clearer view and better comprehension of such 
small changes as may occur nearer us, in the small 
corner of the world in which we are called to live and 
move. We shall know this corner better for the 
journey we have taken into distant lands where we 
had no concern. 

1"" But what we must aim at is not so much to ascertain 
resemblances and differences, as to discover similarities 

Lhidden under apparent discrepancies. The individual 
rules appear at first discordant, but on looking closer 
we can generally detect a resemblance ; though differ- 
ing in matter, they approximate in form and in the 
order of their parts. When we examine them from 
this point of view, we shall see them widen and tend 
to embrace everything. This is what gives a value to 
certain facts that come to complete a whole, and 
show that it is the faithful image of other known 

I cannot dwell further on this point, but these few 
words will suffice to show that the scientist does not 
make a random selection of the facts to be observed. 
He does not count lady-birds, as Tolstoi says, because 
the number of these insects, interesting as they are, is 
subject to capricious variations. He tries to condense 
a great deal of experience and a great deal of thought 
into a small volume, and that is why a little book on 
physics contains so many past experiments, and a 


thousand times as many possible ones, whose results 
are known in advance. 

But so far we have only considered one side of the 
question. The scientist does not study nature because 
it is useful to do so. He studies it because he takes 
pleasure in it, and he takes pleasure in it because it is 
beautiful. If nature were not beautiful it would not be 
worth knowing, and life would not be worth living. I 
am not speaking, of course, of that beauty which 
strikes the senses, of the beauty of qualities and ap- 
pearances. I am far from despising this, but it has 
nothing to do with science. What I mean is that 
more intimate beauty which comes from the harmo- 
nious order of its parts, and which a pure intelligence 
can grasp. It is this that gives a body a skeleton, 
so to speak, to the shimmering visions that flatter 
our senses, and without this support the beauty 
of these fleeting dreams would be imperfect, because 
it would be indefinite and ever elusive. Intellectual 
beauty, on the contrary, is self-sufficing, and it is for 
it, more perhaps than for the future good of humanity, 
that the scientist condemns himself to long and painful 

It is, then, the search for this special beauty, the 
sense of the harmony of the world, that makes us 
select the facts best suited to contribute to this har- 
mony ; just as the artist selects those features of his 
sitter which complete the portrait and give it character 
and life. And there is no fear that this instinctive 
and unacknowledged preoccupation will divert the 
scientist from the search for truth. We may dream 
of a harmonious world, but how far it will fall short 
of the real world ! The Greeks, the greatest artists 


that ever were, constructed a heaven for themselves ; 
how poor a thing it is beside the heaven as we know it ! 

It is because simpHcity and vastness are both beau- 
tiful that we seek by preference simple facts and vast 
facts ; that we take delight, now in following the giant 
courses of the stars, now in scrutinizing with a micro- 
scope that prodigious smallness which is also a vastness, 
and now in seeking in geological ages the traces of a 
past that attracts us because of its remoteness. 

Thus we see that care for the beautiful leads us to 
the same selection as care for the useful. Similarly 
economy of thought, that economy of effort which, 
according to Mach, is the constant tendency of science, 
is a source of beauty as well as a practical advantage. 
The buildings we admire are those in which the archi- 
tect has succeeded in proportioning the means to the 
end, in which the columns seem to carry the burdens 
imposed on them lightly and without effort, like the 
graceful caryatids of the Erechtheum. 

Whence comes this concordance? Is it merely 
that things which seem to us beautiful are those 
which are best adapted to our intelligence, and that 
consequently they are at the same time the tools that 
intelligence knows best how to handle ? Or is it due 
rather to evolution and natural selection ? Have the 
peoples whose ideal conformed best to their own in- 
terests, properly understood, exterminated the others 
and taken their place? One and all pursued their 
ideal without considering the consequences, but while 
this pursuit led some to their destruction, it gave 
empire to others. We are tempted to believe this, 
for if the Greeks triumphed over the barbarians, and 
if Europe, heir of the thought of the Greeks, dominates 


the world, it is due to the fact that the savages loved 
garish colours and the blatant noise of the drum, which 
appealed to their senses, while the Greeks loved the 
intellectual beauty hidden behind sensible beauty, and 
that it is this beauty which gives certainty and strength 
to the intelligence. 

No doubt Tolstoi would be horrified at such a 
triumph, and he would refuse to admit that it could 
be truly useful. But this disinterested pursuit of truth 
for its own beauty is also wholesome, and can make 
men better. I know very well there are disappoint- 
ments, that the thinker does not always find the 
serenity he should, and even that some scientists have 
thoroughly bad tempers. 

Must we therefore say that science should be 
abandoned, and morality alone be studied ? Does 
any one suppose that moralists themselves are entirely 
above reproach when they have come down from the 



If we wish to foresee the future of mathematics, our 
proper course is to study the history and present 
condition of the science. 

For us mathematicians, is not this procedure to 
some extent professional ? We are accustomed to 
extrapolation, which is a method of deducing the 
future from the past and the present ; and since we 
are well aware of its limitations, we run no risk of 
deluding ourselves as to the scope of the results it 
gives us. 

In the past there have been prophets of ill. They 
took pleasure in repeating that all problems suscep- 
tible of being solved had already been solved, and that 
after them there would be nothing left but gleanings. 
Happily we are reassured by the example of the 
past. Many times already men have thought that 
they had solved all the problems, or at least that 
they had made an inventory of all that admit of 
solution. And then the meaning of the word solution 
has been extended ; the insoluble problems have 
become the most interesting of all, and other problems 
hitherto undreamed of have presented themselves. 
For the Greeks a good solution was one that em- 


ployed only rule and compass ; later it became one 
obtained by the extraction of radicals, then one in 
which algebraical functions and radicals alone figured. 
Thus the pessimists found themselves continually 
passed over, continually forced to retreat, so that at 
present I verily believe there are none left. 

My intention, therefore, is not to refute them, since 
they are dead. We know very well that mathematics 
will continue to develop, but we have to find out in 
what direction. I shall be told "in all directions," 
and that is partly true ; but if it were altogether true, 
it would become somewhat alarming. Our riches 
would soon become embarrassing, and their accumula- 
tion would soon produce a mass just as impenetrable 
as the unknown truth was to the ignorant. 

The historian and the physicist himself must make 
a selection of facts. The scientist's brain, which is 
only a corner of the universe, will never be able to 
contain the whole universe ; whence it follows that, 
of the innumerable facts offered by nature, we shall 
leave some aside and retain others. The same is 
true, a fortiori, in mathematics. The mathematician 
similarly cannot retain pell-mell all the facts that are 
presented to him, the more so that it is himself— I was 
almost going to say his own caprice — that creates these 
facts. It is he who assembles the elements and con- 
structs a new combination from top to bottom ; it is 
generally not brought to him ready-made by nature. 

No doubt it is sometimes the case that a mathe- 
matician attacks a problem to satisfy some require- 
ment of physics, that the physicist or the engineer 
asks him to make a calculation in view of some par- 
ticular application. Will it be said that we geometri- 


clans are to confine ourselves to waiting for orders, 
and, instead of cultivating our science for our own 
pleasure, to have no other care but that of accom- 
modating ourselves to our clients' tastes? If the only 
object of mathematics is to come to the help of those 
who make a study of nature, it is to them we must 
look for the word of command. Is this the correct 
view of the matter ? Certainly not ; for if we had not 
cultivated the exact sciences for themselves, we should 
never have created the mathematical instrument, and 
when the word of command came from the physicist 
we should have been found without arms. 

Similarly, physicists do not wait to study a phenom- 
enon until some pressing need of material life makes 
it an absolute necessity, and they are quite right. If 
the scientists of the eighteenth century had dis- 
regarded electricity, because it appeared to them 
merely a curiosity having no practical interest, we 
should not have, in the twentieth century, either 
telegraphy or electro-chemistry or electro -traction. 
Physicists forced to select are not guided in their 
selection solely by utility. What method, then, do 
they pursue in making a selection between the dif- 
ferent natural facts ? I have explained this in the 
preceding chapter. The facts that interest them are 
those that may lead to the discovery of a law, those 
• that have an analogy with many other facts and do 
not appear to us as isolated, but as closely grouped 
with others. The isolated fact attracts the attention 
of all, of the layman as well as the scientist. But 
what the true scientist alone can see is the link that 
unites several facts which have a deep but hidden 
analogy. The anecdote of Newton's apple is probably 


not true, but it is symbolical, so we will treat it as if 
it were true. Well, we must suppose that before 
Newton's day many men had seen apples fall, but 
none had been able to draw any conclusion. Facts 
would be barren if there were not minds capable of 
selecting between them and distinguishing those which 
have something hidden behind them and recognizing 
what is hidden — minds which, behind the bare fact, 
can detect the soul of the fact 

In mathematics we do exactly the same thing. Of 
the various elements at our disposal we can form 
millions of different combinations, but any one of 
these combinations, so long as it is isolated, is ab- 
solutely without value ; often we have taken great 
trouble to construct it, but it is of absolutely no use, 
unless it be, perhaps, to supply a subject for an exer- 
cise in secondary schools. It will be quite different 
as soon as this combination takes its place in a class 
of analogous combinations whose analogy we have 
recognized ; we shall then be no longer in presence of 
a fact, but of a law. And then the true discoverer 
will not be the workman who has patiently built up 
some of these combinations, but the man who has 
brought out their relation. The former has only seen 
the bare fact, the latter alone has detected the soul of 
the fact. The invention of a new word will often 
be sufficient to bring out the relation, and the word 
will be creative. The history of science furnishes us 
with a host of examples that are familiar to all. 

The celebrated Viennese philosopher Mach has said 
that the part of science is to effect economy of thought, 
just as a machine effects economy of effort, and this is 
very true. The savage calculates on his fingers, or 


by putting togethei' pebbles. By teaching children the 
multiplication table we save them later on countless 
operations with pebbles. Some one once recognized, 
whether by pebbles or otherwise, that 6 times 7 
are 42, and had the idea of recording the result, and 
that is the reason why we do not need to repeat the 
operation. His time was not wasted even if he was 
only calculating for his own amusement. His opera- 
tion only took him two minutes, but it would have 
taken two million, if a million people had had to 
repeat it after him. ^ 

Thus the importance of a fact is measured by the 
return it gives — that is, by the amount of thought it 
enables us to economize. 

In physics, the facts which give a large return are 
those which take their place in a very general law, 
because they enable us to foresee a very large number 
of others, and it is exactly the same in mathematics. 
Suppose I apply myself to a complicated calculation 
and with much difficulty arrive at a result, I shall 
have gained nothing by my trouble if it has not 
enabled me to foresee the results of other analogous 
calculations, and to direct them with certainty, avoid- 
ing the blind groping with which I had to be con- 
tented the first time. On the contrary, my time will 
not have been lost if this very groping has succeeded 
in revealing to me the profound analogy between the 
problem just dealt with and a much more extensive 
class of other problems ; if it has shown me at once 
their resemblances and their differences ; if, in a word, 
it has enabled me to perceive the possibility of a 
generalization. Then it will not be merely a new 
result that I have acquired, but a new force. 


An algebraical formula which gives us the solution 
of a type of numerical problems, if we finally replace 
the letters by numbers, is the simple example which 
occurs to one's mind at once. Thanks to the formula, 
a single algebraical calculation saves us the trouble of 
a constant repetition of numerical calculations. But 
this is only a rough example : every one feels that 
there are analogies which cannot be expressed by a 
formula, and that they are the most valuable. 

If a new result is to have any value, it must unite 
elements long since known, but till then scattered 
and seemingly foreign to each other, and suddenly 
introduce order where the appearance of disorder 
reigned. Then it enables us to see at a glance each 
of these elements in the place it occupies in the whole. 
Not only is the new fact valuable on its own account, 
but it alone gives a value to the old facts it unites. 
Our mind is frail as our senses are ; it would lose 
itself in the complexity of the world if that complexity 
were not harmonious ; like the short-sighted, it would 
only see the details, and would be obliged to forget 
each of these details before examining the next, 
because it would be incapable of taking in the whole. 
The only facts worthy of our attention are those 
which introduce order into this complexity and so 
make it accessible to us. 

Mathematicians attach a great importance to the 
elegance of their methods and of their results, and 
this is not mere dilettantism. What is it that gives 
us the feeling of elegance in a solution or a demonstra- 
tion ? It is the harmony of the different parts, their 
symmetry, and their happy adjustment ; it is, in a 
word, all that introduces order, all that gives them 


unity, that enables us to obtain a clear comprehension 
of the whole as well as of the parts. But that is 
also precisely what causes it to give a large return ; 
and in fact the more we see this whole clearly and 
at a single glance, the better we shall perceive the 
analogies with other neighbouring objects, and con- 
sequently the better chance we shall have of guessing 
the possible generalizations. Elegance may result 
from the feeling of surprise caused by the un- 
looked-for occurrence together of objects not habitu- 
ally associated. In this, again, it is fruitful, since it 
thus discloses relations till then unrecognized. It is 
also fruitful even when it only results from the con- 
trast between the simplicity of the means and the 
complexity of the problem presented, for it then causes 
us to reflect on the reason for this contrast, and gener- 
ally shows us that this reason is not chance, but is to 
be found in some unsuspected law. Briefly stated, the 
sentiment of mathematical elegance is nothing but the 
satisfaction due to some conformity between the solu- 
tion we wish to discover and the necessities of our 
mind, and it is on account of this very conformity 
that the solution can be an instrument for us. This 
aesthetic satisfaction is consequently connected with 
the economy of thought. Again the comparison with 
the Erechtheum occurs to me, but I do not wish to 
serve it up too often. 

It is for the same reason that, when a somewhat 
lengthy calculation has conducted us to some simple 
and striking result, we are not satisfied until we have 
shown that we might have foreseen, if not the whole 
result, at least its most characteristic features. Why 
is this ? What is it that prevents our being contented 


with a calculation which has taught us apparently all 
that we wished to know? The reason is that, in 
analogous cases, the lengthy calculation might not be 
able to be used again, while this is not true of the 
reasoning, often semi-intuitive, which might have 
enabled us to foresee the result. This reasoning 
being short, we can see all the parts at a single glance, 
so that we perceive immediately what must be changed 
to adapt it to all the problems of a similar nature 
that may be presented. And since it enables us to 
foresee whether the solution of these problems will 
be simple, it shows us at least whether the calculation 
is worth undertaking. 

What I have just said is sufficient to show how vain 
it would be to attempt to replace the mathematician's 
free initiative by a mechanical process of any kind. 
In order to obtain a result having any real value, it 
is not enough to grind out calculations, or to have 
a machine for putting things in order : it is not order 
only, but unexpected order, that has a value. A 
machine can take hold of the bare fact, but the soul 
of the fact will always escape it. 

Since the middle of last century, mathematicians 
have become more and more anxious to attain to 
absolute exactness. They are quite right, and this 
tendency will become more and more marked. In 
mathematics, exactness is not everything, but without 
it there is nothing : a demonstration which lacks 
exactness is nothing at all. This is a truth that I 
think no one will dispute, but if it is taken too 
literally it leads us to the conclusion that before 1820, 
for instance, there was no such thing as mathematics, 
and this is clearly an exaggeration. The geometri- 


cians of that day were willing to assume what we 
explain by prolix dissertations. This does not mean 
that they did not see it at all, but they passed it 
over too hastily, and, in order to see it clearly, they 
would have had to take the trouble to state it. 

Only, is it always necessary to state it so many 
times ? Those who were the first to pay special 
attention to exactness have given us reasonings that 
we may attempt to imitate ; but if the demonstrations 
of the future are to be constructed on this model, 
mathematical works will become exceedingly long, 
and if I dread length, it is not only because I am 
afraid of the congestion of our libraries, but because 
I fear that as they grow in length our demonstrations 
will lose that appearance of harmony which plays such 
a useful part, as I have just explained. 

It is economy of thought that we should aim at, 
and therefore it is not sufficient to give models to 
be copied. We must enable those that come after 
us to do without the models, and not to repeat a 
previous reasoning, but summarize it in a few lines. 
And this has already been done successfully in certain 
cases. For instance, there was a whole class of reason- 
ings that resembled each other, and were found every- 
where ; they were perfectly exact, but they were long. 
One day some one thought of the term " uniformity of 
convergence," and this term alone made them useless ; 
it was no longer necessary to repeat them, since they 
could now be assumed. Thus the hair-splitters can 
render us a double service, first by teaching us to 
do as they do if necessary, but more especially by 
enabling us as often as possible not to do as they 

do, and yet make no sacrifice of exactness. 
(1.777) 3 


One example has just shown us the importance 
of terms in mathematics ; but I could quote many 
others. It is hardly possible to believe what economy 
of thought, as Mach used to say, can be effected by 
a well-chosen term. I think I have already said 
somewhere that mathematics is the art of giving the 
same name to different things. It is enough that 
these things, though differing in matter, should be 
similar in form, to permit of their being, so to speak, 
run in the same mould. When language has been 
well chosen, one is astonished to find that all demon- 
strations made for a known object apply immediately 
to many new objects : nothing requires to be changed, 
not even the terms, since the names have become the 

A well-chosen term is very often sufficient to remove 
the exceptions permitted by the rules as stated in the 
old phraseology. This accounts for the invention of 
negative quantities, imaginary quantities, decimals to 
infinity, and I know not what else. And we must 
never forget that exceptions are pernicious, because 
they conceal laws. 

This is one of the characteristics by which we re- 
cognize facts which give a great return : they are the 
facts which permit of these happy innovations of 
language. The bare fact, then, has sometimes no great 
interest : it may have been noted many times without 
rendering any great service to science ; it only acquires 
a value when some more careful thinker perceives the 
connexion it brings out, and symbolizes it by a term. 

The physicists also proceed in exactly the same 
way. They have invented the term " energy," and the 
term has been enormously fruitful, because it also 


creates a law by eliminating exceptions ; because it 
gives the same name to things which differ in matter, 
but are similar in form. 

Among the terms which have exercised the most 
happy influence I would note "group" and "invariable." 
They have enabled us to perceive the essence of many 
mathematical reasonings, and have shown us in how 
many cases the old mathematicians were dealing with 
groups without knowing it, and how, believing them- 
selves far removed from each other, they suddenly 
found themselves close together without understanding 

To-day we should say that they had been examining 
isomorphic groups. We now know that, in a group, the 
matter is of little interest, that the form only is of 
importance, and that when we are well acquainted 
with one group, we know by that very fact all the 
isomorphic groups. Thanks to the terms " group " and 
"isomorphism," which sum up this subtle rule in a 
few syllables, and make it readily familiar to all minds, 
the passage is immediate, and can be made without 
expending any effort of thinking. The idea of group 
is, moreover, connected with that of transformation. 
Why do we attach so much value to the discovery 
of a new transformation ? It is because, from a single 
theorem, it enables us to draw ten or twenty others. 
It has the same value as a zero added to the right 
of a whole number. 

This is what has determined the direction of the 
movement of mathematical science up to the present, 
and it is also most certainly what will determine it 
in the future. But the nature of the problems which 
present themselves contributes to it in an equal degree. 


We cannot forget what our aim should be, and in my 
opinion this aim is a double one. Our science borders 
on both philosophy and physics, and it is for these 
two neighboMrs that we must work. And so we have 
always seen, and we shall still see, mathematicians 
advancing in two opposite directions. 

On the one side, mathematical science must reflect 
upon itself, and this is useful because reflecting upon 
itself is reflecting upon the human mind which has 
created it ; the more so because, of all its creations, 
mathematics is the one for which it has borrowed 
least from outside. This is the reason for the utility 
of certain mathematical speculations, such as those 
which have in view the study of postulates, of un- 
usual geometries, of functions with strange behaviour. 
The more these speculations depart from the most 
ordinary conceptions, and, consequently, from nature 
and applications to natural problems, the better will 
they show us what the human mind can do when it 
is more and more withdrawn from the tyranny of 
the exterior world ; the better, consequently, will they 
make us know this mind itself 

But it is to the opposite side, to the side of nature, 
that we must direct our main forces. 

There we meet the physicist or the engineer, who 
says, " Will you integrate this differential equation for 
me ; I shall need it within a week for a piece of 
construction work that has to be completed by a 
certain date ? " " This equation," we answer, " is not 
included in one of the types that can be integrated, 
of which you know there are not very many." " Yes, 
I know ; but, then, what good are you ? " More often 
than not a mutual understanding is sufficient. The 


engineer does not really require the integral in finite 
terms, he only requires to know the general behaviour 
of the integral function, or he merely wants a certain 
figure which would be easily deduced from this in- 
tegral if we knew it. Ordinarily we do not know 
it, but we could calculate the figure without it, if we 
knew just what figure and what degree of exactness 
the engineer required. 

Formerly an equation was not considered to have 
been solved until the solution had been expressed 
by means of a finite number of known functions. 
But this is impossible in about ninety-nine cases 
out of a hundred. What we can always do, or rather 
what we should always try to do, is to solve the 
problem qualitatively, so to speak — that is, to try to 
know approximately the general form of the curve 
which represents the unknown function. 

It then remains to find the exact solution of the 
problem. But if the unknown cannot be determined 
by a finite calculation, we can always represent it 
by an infinite converging series which enables us to 
calculate it. Can this be regarded as a true solu- 
tion ? The story goes that Newton once communi- 
cated to Leibnitz an anagram somewhat like the 
following : aaaaabbbeeeeii, etc. Naturally, Leibnitz 
did not understand it at all, but we who have the 
key know that the anagram, translated into modern 
phraseology, means, " I know how to integrate all 
differential equations," and we are tempted to make 
the comment that Newton was either exceedingly 
fortunate or that he had very singular illusions. 
What he meant to say was simply that he could 
form (by means of indeterminate coefficients) a 


series of powers formally satisfying the equation 

To-day a similar solution would no longer satisfy 
us, for two reasons — because the convergence is too 
slow, and because the terms succeed one another 
without obeying any law. On the other hand the 
series 9 appears to us to leave nothing to be desired, 
first, because it converges very rapidly (this is for 
the practical man who wants his number as quickly 
as possible), and secondly, because we perceive at a 
glance the law of the terms, which satisfies the 
esthetic requirements of the theorist. 

There are, therefore, no longer some problems 
solved and others unsolved, there are only problems 
more or less solved, according as this is accomplished 
by a series of more or less rapid convergence or 
regulated by a more or less harmonious law. Never- 
theless an imperfect solution may happen to lead 
us towards a better one. 

Sometimes the series is of such slow convergence 
that the calculation is impracticable, and we have 
only succeeded in demonstrating the possibility of 
the problem. The engineer considers this absurd, 
and he is right, since it will not help him to com- 
plete his construction within the time allowed. He 
doesn't trouble himself with the question whether it 
will be of use to the engineers of the twenty-second 
century. We think differently, and we are sometimes 
more pleased at having economized a day's work 
for our grandchildren than an hour for our contem- 

Sometimes by groping, so to speak, empirically, 
we arrive at a formula that is sufficiently convergent. 


What more would you have? says the engineer; and 
yet, in spite of everything, we are not satisfied, for 
we should have liked to be able to predict the con- 
vergence. And why? Because if we had known 
how to predict it in the one case, we should know 
how to predict it in another. We have been success- 
ful, it is true, but that is little in our eyes if we have 
no real hope of repeating our success. 

In proportion as the science develops, it becomes 
more difficult to take it in in its entirety. Then an 
attempt is made to cut it in pieces and to be satisfied 
with one of these pieces — in a word, to specialize. Too 
great a movement in this direction would constitute 
a serious obstacle to the progress of the science. As 
I have said, it is by unexpected concurrences between 
its different parts that it can make progress. Too 
much specializing would prohibit these concurrences. 
Let us hope that congresses, such as those of Heidel- 
berg and Rome, by putting us in touch with each 
other, will open up a view of our neighbours' territory, 
and force us to compare it with our own, and so 
escape in a measure from our own little village. In 
this way they will be the best remedy against the 
danger I have just noted. 

But I have delayed too long over generalities ; it 
is time to enter into details. 

Let us review the different particular sciences which 
go to make up mathematics ; let us see what each of 
them has done, in what direction it is tending, and 
what we may expect of it. If the preceding views 
are correct, we should see that the great progress of 
the past has been made when two of these sciences 
have been brought into conjunction, when men have 


become aware of the similarity of their form in spite 
of the dissimilarity of their matter, when they have 
modelled themselves upon each other in such a way 
that each could profit by the triumphs of the other. 
At the same time we should look to concurrences of 
a similar nature for progress in the future. 


The progress of arithmetic has been much slower 
than that of algebra and analysis, and it is easy to 
understand the reason. The feeling of continuity is 
a precious guide which fails the arithmetician. 
Every whole number is separated from the rest, and 
has, so to speak, its own individuality ; each of them 
is a sort of exception, and that is the reason why 
general theorems will always be less common in 
the theory of numbers, and also why those that do 
exist will be more hidden and will longer escape 

If arithmetic is backward as compared with algebra 
and analysis, the best thing for it to do is to try to 
model itself on these sciences, in order to profit by 
their advance. The arithmetician then should be 
guided by the analogies with algebra. These analo- 
gies are numerous, and if in many cases they have 
not yet been studied sufficiently closely to become 
serviceable, they have at least been long foreshadowed, 
and the very language of the two sciences shows 
that they have been perceived. Thus we speak of 
transcendental numbers, and so become aware of 
the fact that the future classification of these numbers 
has already a model in the classification of transcen- 
dental functions. However, it is not yet very clear 


how we are to pass from one classification to the 
other ; but if it were clear it would be already done, 
and would no longer be the work of the future. 

The first example that comes to my mind is the 
theory of congruents, in which we find a perfect 
parallelism with that of algebraic equations. We 
shall certainly succeed in completing this parallelism, 
which must exist, for instance, between the theory of 
algebraic curves and that of congruents with two 
variables. When the problems relating to congruents 
with several variables have been solved, we shall have 
made the first step towards the solution of many ques- 
tions of indeterminate analysis. 


The theory of algebraic equations will long continue 
to attract the attention of geometricians, the sides by 
which it may be approached being so numerous and 
so different 

It must not be supposed that algebra is finished 
because it furnishes rules for forming all possible 
combinations ; it still remains to find interesting com- 
binations, those that satisfy such and such conditions. 
Thus there will be built up a kind of indeterminate 
analysis, in which the unknown quantities will no 
longer be whole numbers but polynomials. So this 
time it is algebra that will model itself on arithmetic, 
being guided by the analogy of the whole number, 
either with the whole polynomial with indefinite 
coefificients, or with the whole polynomial with whole 



It would seem that geometry can contain nothing 
that is not already contained in algebra or analysis, and 
that geometric facts are nothing but the facts of algebra 
or analysis expressed in another language. It might 
be supposed, then, that after the review that has just 
been made, there would be nothing left to say having 
any special bearing on geometry. But this would 
imply a failure to recognize the great importance of a 
well-formed language, or to understand what is added 
to things themselves by the method of expressing, and 
consequently of grouping, those things. 

To begin with, geometric considerations lead us to 
set ourselves new problems. These are certainly, if 
you will, analytical problems, but they are problems 
we should never have set ourselves on the score of 
analysis. Analysis, however, profits by them, as it 
profits by those it is obliged to solve in order to 
satisfy the requirements of physics. 

One great advantage of geometry lies precisely in 
the fact that the senses can come to the assistance of 
the intellect, and help to determine the road to be 
followed, and many minds prefer to reduce the 
problems of analysis to geometric form. Unfortu- 
nately our senses cannot carry us very far, and they 
leave us in the lurch as soon as we wish to pass 
outside the three classical dimensions. Does this 
mean that when we have left this restricted domain 
in which they would seem to wish to imprison us, we 
must no longer count on anything but pure analysis, 
and that all geometry of more than three dimensions 
is vain and without object ? In the generation which 


preceded ours, the greatest masters would have an- 
swered "Yes." To-day we are so familiar with this 
notion that we can speak of it, even in a university 
course, without exciting too much astonishment. 

But of what use can it be ? This is easy to see. In 
the first place it gives us a very convenient language, 
which expresses in very concise terms what the ordi- 
nary language of analysis would state in long-winded 
phrases. More than that, this language causes us to 
give the same name to things which resemble one 
another, and states analogies which it does not allow 
us to forget. It thus enables us still to find our way 
in that space which is too great for us, by calling to 
our mind continually the visible space, which is only 
an imperfect image of it, no doubt, but still an image. 
Here again, as in all the preceding examples, it is 
the analogy with what is simple that enables us to 
understand what is complex. 

This geometry of more than three dimensions is 
not a simple analytical geometry, it is not purely 
quantitative, but also qualitative, and it is principally 
on this ground that it becomes interesting. There is a 
science called Geometry of Position, which has for its 
object the study of the relations of position of the 
different elements of a figure, after eliminating their 
magnitudes. This geometry is purely qualitative ; its 
theorems would remain true if the figures, instead of 
being exact, were rudely imitated by a child. We can 
also construct a Geometry of Position of more than 
three dimensions. The importance of Geometry of 
Position is immense, and I cannot insist upon it too 
much ; what Riemann, one of its principal creators, 
has gained from it would be sufficient to demonstrate 


this. We must succeed in constructing it completely 
in the higher spaces, and we shall then have an instru- 
ment which will enable us really to see into hyperspace 
and to supplement our senses. 

The problems of Geometry of Position would perhaps 
not have presented themselves if only the language of 
analysis had been used. Or rather I am wrong, for 
they would certainly have presented themselves, since 
their solution is necessary for a host of questions of 
analysis, but they would have presented themselves 
isolated, one after the other, and without our being 
able to perceive their common link. 


I have spoken above of the need we have of 
returning continually to the first principles of our 
science, and of the advantage of this process to the 
study of the human mind. It is this need which has 
inspired two attempts which have held a very great 
place in the most recent history of mathematics. The 
first is Cantorism, and the services it has rendered to 
the science are well known. Cantor introduced into 
the science a new method of considering mathematical 
infinity, and I shall have occasion to speak of it again 
in Book tl., chapter iii. One of the characteristic 
features of Cantorism is that, instead of rising to the 
general by erecting more and more complicated con- 
structions, and defining by construction, it starts with 
the genus supreinum and only defines, as the scholastics 
would have said, per genus proximum et differe?ttiam 
specificam. Hence the horror he has sometimes in- 
spired in certain minds, such as Hermitte's, whose 
favourite idea was to compare the mathematical with 


the natural sciences. For the greater number of us 
these prejudices had been dissipated, but it has come 
about that we have run against certain paradoxes and 
apparent contradictions, which would have rejoiced 
the heart of Zeno of Elea and the school of Megara. 
Then began the business of searching for a remedy, 
each man his own way. For my part I think, and I 
am not alone in so thinking, that the important thing 
is never to introduce any entities but such as can be 
completely defined in a finite number of words. What- 
ever be the remedy adopted, we can promise ourselves 
the joy of the doctor called in to follow a fine patho- 
logical case. 

The Search for Postulates. 

Attempts have been made, from another point of 
view, to enumerate the axioms and postulates more 
or less concealed which form the foundation of the 
different mathematical theories, and in this direction 
Mr. Hilbert has obtained the most brilliant results. 
It seems at first that this domain must be strictly 
limited, and that there will be nothing more to do 
when the inventory has been completed, which cannot 
be long. But when everything has been enumerated, 
there will be many ways of classifying it all. A good 
librarian always finds work to do, and each new classi- 
fication will be instructive for the philosopher. 

I here close this review, which I cannot dream of 
making complete. I think that these examples will 
have been sufficient to show the mechanism by which 
the mathematical sciences have progressed in the past, 
and the direction in which they must advance in the 



The genesis of mathematical discovery is a problem 
which must inspire the psychologist with the keenest 
interest. For this is the process in which the human 
mind seems to borrow least from the exterior world, 
in which it acts, or appears to act, only by itself and 
on itself, so that by studying the process of geometric 
thought we may hope to arrive at what is most 
essential in the human mind. 

This has long been understood, and a few months 
ago a review called l' Enseignement Mathematique, 
edited by MM. Laisant and Fehr, instituted an en- 
quiry into the habits of mind and methods of work 
of different mathematicians. I had outlined the 
principal features of this article when the results of 
the enquiry were published, so that I have hardly been 
able to make any use of them, and I will content 
myself with saying that the majority of the evidence 
confirms my conclusions. I do not say there is 
unanimity, for on an appeal to universal suffrage we 
cannot hope to obtain unanimity. 

One first fact must astonish us, or rather would 
astonish us if we were not too much accustomed to 
it. How does it happen that there are people who 


do not understand mathematics? If the science 
invokes only the rules of logic, those accepted by 
all well-formed minds, if its evidence is founded on 
principles that are common to all men, and that none 
but a madman would attempt to deny, how does it 
happen that there are so many people who are 
entirely impervious to it ? 

There is nothing mysterious in the fact that every 
one is not capable of discovery. That every one 
should not be able to retain a demonstration he has 
once learnt is still comprehensible. But what does 
seem most surprising, when we consider it, is that 
any one should be unable to understand a mathe- 
matical argument at the very moment it is stated to 
him. And yet those who can only follow the argu- 
ment with difficulty are in a majority ; this is incon- 
testable, and the experience of teachers of secondary 
education will certainly not contradict me. 

And still further, how is error possible in mathe- 
matics ? A healthy intellect should not be guilty 
of any error in logic, and yet there are very keen 
minds which will not make a false step in a short 
argument such as those we have to make in the 
ordinary actions of life, which yet are incapable of 
following or repeating without error the demonstra- 
tions of mathematics which are longer, but which 
are, after all, only accumulations of short arguments 
exactly analogous to those they make so easily. Is it 
necessary to add that mathematicians themselves are 
not infallible? 

The answer appears to me obvious. Imagine a 
long series of syllogisms in which the conclusions of 
those that precede form the premises of those that 


follow. We shall be capable of grasping each of the 
syllogisms, and it is not in the passage from premises 
to conclusion that we are in danger of going astray. 
But between the moment when we meet a proposition 
for the first time as the conclusion of one syllogism, 
and the moment when we find it once more as the 
premise of another syllogism, much time will some- 
times have elapsed, and we shall have unfolded many 
links of the chain ; accordingly it may well happen 
that we shall have forgotten it, or, what is more serious, 
forgotten its meaning. So we may chance to replace 
it by a somewhat different proposition, or to preserve 
the same statement but give it a slightly different 
meaning, and thus we are in danger of falling into 

A mathematician must often use a rule, and, natur- 
ally, he begins by demonstrating the rule. At the 
moment the demonstration is quite fresh in his 
memory he understands perfectly its meaning and 
significance, and he is in no danger of changing it. 
But later on he commits it to memory, and only 
applies it in a mechanical way, and then, if his 
memory fails him, he may apply it wrongly. It is 
thus, to take a simple and almost vulgar example, 
that we sometimes make mistakes in calculation, 
because we have forgotten our multiplication table. 

On this view special aptitude for mathematics 
would be due to nothing but a very certain memory 
or a tremendous power of attention. It would be a 
quality analogous to that of the whist player who 
can remember the cards played, or, to rise a step 
higher, to that of the chess player who can picture 
a very great number of combinations and retain them 


in his memory. Every good mathematician should 
also be a good chess player and vice versa, and 
similarly he should be a good numerical calculator. 
Certainly this sometimes happens, and thus Gauss 
was at once a geometrician of genius and a very 
precocious and very certain calculator. 

But there are exceptions, or rather I am wrong, 
for I cannot call them exceptions, otherwise the excep- 
tions would be more numerous than the cases of con- 
formity with the rule. On the contrary, it was Gauss 
who was an exception. As for myself, I must confess 
I am absolutely incapable of doing an addition sum 
without a mistake. Similarly I should be a very bad 
chess player. I could easily calculate that by playing 
in a certain way I should be exposed to such and 
such a danger ; I should then review many other 
moves, which I should reject for other reasons, and 
I should end by making the move I first examined, 
having forgotten in the interval the danger I had 

In a word, my memory is not bad, but it would be 
insufficient to make me a good chess player. Why, 
then, does it not fail me in a difficult mathematical 
argument in which the majority of chess players 
would be lost ? Clearly because it is guided by the 
general trend of the argument. A mathematical 
demonstration is not a simple juxtaposition of syl- 
logisms ; it consists of syllogisms placed in a certain 
order, and the order in which these elements are 
placed is much more important than the elements 
themselves. If I have the feeling, so to speak the 
intuition, of this order, so that I can perceive the 

whole of the argument at a glance, I need no longer 
(1,777) 4 


be afraid of forgetting one of the elements ; each of 
them will place itself naturally in the position pre- 
pared for it, without my having to make any effort 
of memory. 

It seems to me, then, as I repeat an argument I 
have learnt, that I could have discovered it. This 
is often only an illusion ; but even then, even if I am 
not clever enough to create for myself, I rediscover 
it myself as I repeat it. 

We can understand that this feeling, this intuition 
of mathematical order, which enables us to guess 
hidden harmonies and relations, cannot belong to 
every one. Some have neither this delicate feeling 
that is difficult to define, nor a power of memory and 
attention above the common, and so they are abso- 
lutely incapable of understanding even the first steps 
of higher mathematics. This applies to the majority 
of people. Others have the feeling only in a slight 
degree, but they are gifted with an uncommon 
memory and a great capacity for attention. They 
learn the details one after the other by heart, they 
can understand mathemathics and sometimes apply 
them, but they are not in a condition to create. 
Lastly, others possess the special intuition I have 
spoken of more or less highly developed, and they 
can not only understand mathematics, even though 
their memory is in no way extraordinary, but they 
can become creators, and seek to make discovery 
with more or less chance of success, according as their 
intuition is more or less developed. 

What, in fact, is mathematical discovery? It does 
not consist in making new combinations with mathe- 
matical entities that are already known. That can 


be done by any one, and the combinations that could 
be so formed would be infinite in number, and the 
greater part of them would be absolutely devoid of 
interest. Discovery consists precisely in not con- 
structing useless combinations, but in constructing 
those that are useful, which are an infinitely small 
minority. Discovery is discernment, selection. 

How this selection is to be made I have explained 
above. Mathematical facts worthy of being studied 
are those which, by their analogy with other facts, 
are capable of conducting us to the knowledge of a 
mathematical law, in the same way that experimental 
facts conduct us to the knowledge of a physical law. 
They are those which reveal unsuspected relations 
between other facts, long since known, but wrongly 
believed to be unrelated to each other. 

Among the combinations we choose, the most fruit- 
ful are often those which are formed of elements 
borrowed from widely separated domains. I do not 
mean to say that for discovery it is sufficient to bring 
together objects that are as incongruous as possible. 
The greater part of the combinations so formed would 
be entirely fruitless, but some among them, though 
very rare, are the most fruitful of all. 

Discovery, as I have said, is selection. But this is 
perhaps not quite the right word. It suggests a pur- 
chaser who has been shown a large number of samples, 
and examines them one after the other in order to 
make his selection. In our case the samples would be 
so numerous that a whole life would not give sufficient 
time to examine them. Things do not happen in this 
way. Unfruitful combinations do not so much as 
present themselves to the mind of the discoverer. In 


the field of his consciousness there never appear any 
but really useful combinations, and some that he 
rejects, which, however, partake to some extent of 
the character of useful combinations. Everything 
happens as if the discoverer were a secondary examiner 
who had only to interrogate candidates declared eli- 
gible after passing a preliminary test. 

But what I have said up to now is only what can 
be observed or inferred by reading the works of 
geometricians, provided they are read with some 

It is time to penetrate further, and to see what 
happens in the very soul of the mathematician. For 
this purpose I think I cannot do better than recount 
my personal recollections. Only I am going to confine 
myself to relating how I wrote my first treatise on 
Fuchsian functions. I must apologize, for I am going 
to introduce some technical expressions, but they need 
not alarm the reader, for he has no need to under- 
stand them. I shall say, for instance, that I found the 
demonstration of such and such a theorem under such 
and such circumstances ; the theorem will have a 
barbarous name that many will not know, but that 
is of no importance. What is interesting for the 
psychologist is not the theorem but the circumstances. 

For a fortnight I had been attempting to prove 
that there could not be any function analogous to 
what I have since called Fuchsian functions. I was at 
that time very ignorant. Every day I sat down at my 
table and spent an hour or two trying a great number 
of combinations, and I arrived at no result. One 
night I took some black coffee, contrary to my custom, 
and was unable to sleep. A host of ideas kept surging 


in my head ; I could almost feel then jostling one 
another, until two of them coalesced, so to speak, to 
form a stable combination. When morning came, I 
had established the existence of one class of Fuchsian 
functions, those that are derived from the hyper- 
geometric series. I had only to verify the results, 
which only took a few hours. 

Then I wished to represent these functions by the 
quotient of two series. This idea was perfectly con- 
scious and deliberate ; I was guided by the analogy 
with elliptical functions. I asked myself what must 
be the properties of these series, if they existed, and 
I succeeded without difficulty in forming the series 
that I have called Theta-Fuchsian. 

At this moment I left Caen, where I was then living, 
to take part in a geological conference arranged by 
the School of Mines. The incidents of the journey 
made me forget my mathematical work. When we 
arrived at Coutances, we got into a break to go 
for a drive, and, just as I put my foot on the 
step, the idea came to me, though nothing in my 
former thoughts seemed to have prepared me for it, 
that the transformations I had used to define Fuchsian 
functions were identical with those of non-Euclidian 
geometry. I made no verification, and had no time to 
do so, since I took up the conversation again as soon 
as I had sat down in the break, but I felt absolute 
certainty at once. When I got back to Caen I verified 
the result at my leisure to satisfy my conscience. 

I then began to study arithmetical questions without 
any great apparent result, and without suspecting that 
they could have the least connexion with my previous 
researches. Disgusted at my want of success, I went 


away to spend a few days at the seaside, and 
thought of entirely different things. One day, as I 
was walking on the cliff, the idea came to me, again 
with the same characteristics of conciseness, sudden- 
ness, and immediate certainty, that arithmetical trans- 
formations of indefinite ternary quadratic forms are 
identical with those of non-Euclidian geometry. 

Returning to Caen, I reflected on this result and 
deduced its consequences. The example of quadratic 
forms showed me that there are Fuchsian groups 
other than those which correspond with the hyper- 
geometric series ; I saw that I could apply to them 
the theory of the Theta-Fuchsian series, and that, 
consequently, there are Fuchsian functions other than 
those which are derived from the hypergeometric series, 
the only ones I knew up to that time. Naturally, I 
proposed to form all these functions. I laid siege 
to them systematically and captured all the outworks 
one after the other. There was one, however, which 
still held out, whose fall would carry with it that of the 
central fortress. But all my efforts were of no avail at 
first, except to make me better understand the difficulty, 
which was already something. All this work was per- 
fectly conscious. 

Thereupon I left for Mont-Valerien, where I had 
to serve my time in the army, and so my mind was 
preoccupied with very different matters. One day, as 
I was crossing the street, the solution of the difficulty 
which had brought me to a standstill came to me 
all at once. I did not try to fathom it immediately, 
and it was only after my service was finished that 
I returned to the question. I had all the elements, 
and had only to assemble and arrange them. Accord- 


ingly I composed my definitive treatise at a sitting 
and without any difficulty. 

It is useless to multiply examples, and I will con- 
tent myself with this one alone. As regards my other 
researches, the accounts I should give would be exactly 
similar, and the observations related by other mathe- 
maticians in the enquiry of V Enseigtiement Math'e- 
niatique would only confirm them. 

One is at once struck by these appearances of 
sudden illumination, obvious indications of a long 
course of previous unconscious work. The part played 
by this unconscious work in mathematical discovery 
seems to me indisputable, and we shall find traces 
of it in other cases where it is less evident. Often 
when a man is working at a difficult question, he 
accomplishes nothing the first time he sets to work. 
Then he takes more or less of a rest, and sits down 
again at his table. During the first half-hour he still 
finds nothing, and then all at once the decisive idea 
presents itself to his mind. We might say that the 
conscious work proved more fruitful because it was 
interrupted and the rest restored force and freshness 
to the mind. But it is more probable that the rest 
was occupied with unconscious work, and that the 
result of this work was afterwards revealed to the 
geometrician exactly as in the cases 1 have quoted, 
except that the revelation, instead of coming to light 
during a walk or a journey, came during a period 
of conscious work, but independently of that work, 
which at most only performs the unlocking process, 
as if it were the spur that excited into conscious form 
the results already acquired during the rest, which till 
then remained unconscious. 


There is another remark to be made regarding 
the conditions of this unconscious work, which is, that 
it is not possible, or in any case not fruitful, unless 
it is first preceded and then followed by a period 
of conscious work. These sudden inspirations are 
never produced (and this is sufficiently proved already 
by the examples I have quoted) except after some 
days of voluntary efforts which appeared absolutely 
fruitless, in which one thought one had accomplished 
nothing, and seemed to be on a totally wrong track. 
These efforts, however, were not as barren as one 
thought ; they set the unconscious machine in motion, 
and without them it would not have worked at all, 
and would not have produced anything. 

The necessity for the second period of conscious 
work can be even more readily understood. It is 
necessary to work out the results of the inspiration, 
to deduce the immediate consequences and put them 
in order and to set out the demonstrations ; but, above 
all, it is necessary to verify them. I have spoken 
of the feeling of absolute certainty which accompanies 
the inspiration ; in the cases quoted this feeling was 
not deceptive, and more often than not this will be 
the case. But we must beware of thinking that this 
is a rule without exceptions. Often the feeling de- 
ceives us without being any less distinct on that 
account, and we only detect it when we attempt to 
establish the demonstration. I have observed this 
fact most notably with regard to ideas that have come 
to me in the morning or at night when I have been 
in bed in a semi-somnolent condition. 

Such are the facts of the case, and they suggest the 
following reflections. The result of all that precedes 


is to show that the unconscious ego, or, as it is called, 
the subliminal ego, plays a most important part 
in mathematical discovery. But the subliminal ego 
is generally thought of as purely automatic. Now we 
have seen that mathematical work is not a simple 
mechanical work, and that it could not be entrusted 
to any machine, whatever the degree of perfection we 
suppose it to have been brought to. It is not merely 
a question of applying certain rules, of manufacturing 
as many combinations as possible according to certain 
fixed laws. The combinations so obtained would 
be extremely numerous, useless, and encumbering. ' 
The real work of the discoverer consists in choosing 
between these combinations with a view to eliminating 
those that are useless, or rather not giving himself 1 
the trouble of making them at all. The rules which ; 
must guide this choice are extremely subtle and 
delicate, and it is practically impossible to state them 
in precise language ; they must be felt rather than for- < 
mulated. Under these conditions, how can we imagine ; 
a sieve capable of applying them mechanically ? 

The following, then, presents itself as a first hypoth- 
esis. The subliminal ego is in no way inferior to the 
conscious ego ; it is not purely automatic ; it is capable 
of discernment ; it has tact and lightness of touch ; 
it can select, and it can divine. More than that, 
it can divine better than the conscious ego, since 
it succeeds where the latter fails. In a word, is not 
the subliminal ego superior to the conscious ego? 
The importance of this question will be readily 
understood. In a recent lecture, M. Boutroux showed 
how it had arisen on entirely different occasions, and 
what consequences would be involved by an answer 


in the affirmative. (See also the same author's 
Science et Religion, pp. 313^/" seq}) 

Are we forced to give this affirmative answer by 
the facts I have just stated ? I confess that, for my 
part, I should be loth to accept it. Let us, then, 
return to the facts, and see if they do not admit of 
some other explanation. 

It is certain that the combinations which present 
themselves to the mind in a kind of sudden illumina- 
tion after a somewhat prolonged period of unconscious 
work are generally useful and fruitful combinations, 
which appear to be the result of a preliminary sifting. 
Does it follow from this that the subliminal ego, 
having divined by a delicate intuition that these 
combinations could be useful, has formed none but 
these, or has it formed a great many others which 
were devoid of interest, and remained unconscious ? 

Under this second aspect, all the combinations are 
formed as a result of the automatic action of the 
subliminal ego, but those only which are interesting 
find their way into the field of consciousness. This, too, 
is most mysterious. How can we explain the fact that, 
of the thousand products of our unconscious activity, 
some are invited to cross the threshold, while others 
remain outside? Is it mere chance that gives them 
this privilege? Evidently not. For instance, of 
all the excitements of our senses, it is only the most 
intense that retain our attention, unless it has been 
directed upon them by other causes. More commonly 
the privileged unconscious phenomena, those that are 
capable of becoming conscious, are those which, 
directly or indirectly, most deeply affect our sen- 


It may appear surprising that sensibility should 
be introduced in connexion with mathematical de- 
monstrations, which, it would seem, can only interest 
the intellect. But not if we bear in mind the feeling 
of mathematical beauty, of the harmony of numbers 
and forms and of geometric elegance. It is a real 
jesthetic feeling that all true mathematicians recognize, 
and this is truly sensibility. 

Now, what are the mathematical entities to which 
we attribute this character of beauty and elegance, 
which are capable of developing in us a kind of 
aesthetic emotion ? Those whose elements are har- 
moniously arranged so that the mind can, without 
effort, take in the whole without neglecting the details. 
This harmony is at once a satisfaction to our aesthetic 
requirements, and an assistance to the mind which 
it supports and guides. At the same time, by setting 
before our eyes a well-ordered whole, it gives us 
a presentiment of a mathematical law. Now, as I 
have said above, the only mathematical facts worthy 
of retaining our attention and capable of being useful 
are those which can make us acquainted with a 
mathematical law. Accordingly we arrive at the 
following conclusion. The useful combinations are 
precisely the most beautiful, I mean those that can 
most charm that special sensibility that all mathe- 
maticians know, but of which laymen are so ignorant 
that they are often tempted to smile at it. 

What follows, then ? Of the very large number of 
combinations which the subliminal ego blindly forms, 
almost all are without interest and without utility. 
But, for that very reason, they are without action on 
the aesthetic sensibility ; the consciousness will never 


know them. A few only are harmonious, and con- 
sequently at once useful and beautiful, and they 
will be capable of affecting the geometrician's special 
sensibility 1 have been speaking of; which, once 
aroused, will direct our attention upon them, and will 
thus give them the opportunity of becoming conscious. 

This is only a hypothesis, and yet there is an 
observation which tends to confirm it. When a 
sudden illumination invades the mathematician's mind, 
it most frequently happens that it does not mislead 
him. But it also happens sometimes, as I have said, 
that it will not stand the test of verification. Well, 
it is to be observed almost always that this false idea, 
if it had been correct, would have flattered our natural 
instinct for mathematical elegance. 

Thus it is this special assthetic sensibility that plays 
the part of the delicate sieve of which I spoke above, 
and this makes it sufficiently clear why the man who 
has it not will never be a real discoverer. 

All the difficulties, however, have not disappeared. 
The conscious ego is strictly limited, but as regards 
the subliminal ego, we do not know its limitations, 
and that is why we are not too loth to suppose 
that in a brief space of time it can form more 
different combinations than could be comprised in 
the whole life of a conscient being. These limitations 
do exist, however. Is it conceivable that it can form 
all the possible combinations, whose number staggers 
the imagination ? Nevertheless this would seem to be 
necessary, for if it produces only a small portion of the 
combinations, and that by chance, there vv^ill be very 
small likelihood of the right one, the one that must be 
selected, being found among them. 


Perhaps we must look for the explanation in that 
period of preliminary conscious work which always 
precedes all fruitful unconscious work. If I may 
be permitted a crude comparison, let us represent the 
future elements of our combinations as something 
resembling Epicurus's hooked atoms. When the mind 
is in complete repose these atoms are immovable ; 
they are, so to speak, attached to the wall. This com- 
plete repose may continue indefinitely without the 
atoms meeting, and, consequently, without the pos- 
sibility of the formation of any combination. 

On the other hand, during a period of apparent 
repose, but of unconscious work, some of them are 
detached from the wall and set in motion. They 
plough through space in all directions, like a swarm 
of gnats, for instance, or, if we prefer a more learned 
comparison, like the gaseous molecules in the kinetic 
theory of gases. Their mutual collisions may then 
produce new combinations. 

What is the part to be played by the preliminary 
conscious work ? Clearly it is to liberate some of 
these atoms, to detach them from the wall and set 
them in motion. We think we have accomplished 
nothing, when we have stirred up the elements in a 
thousand different ways to try to arrange them, and 
have not succeeded in finding a satisfactory arrange- 
ment. But after this agitation imparted to them by 
our will, they do not return to their original repose, 
but continue to circulate freely. 

Now our will did not select them at random, but 
in pursuit of a perfectly definite aim. Those it has 
liberated are not, therefore, chance atoms ; they are 
those from which we may reasonably expect the 


desired solution. The liberated atoms will then 
experience collisions, either with each other, or with 
the atoms that have remained stationary, which 
they will run against in their course. I apologize 
once more. My comparison is very crude, but I 
cannot well see how I could explain my thought 
in any other way. 

However it be, the only combinations that have 
any chance of being formed are those in which one 
at least of the elements is one of the atoms deliber- 
ately selected by our will. Now it is evidently 
among these that what I called just now the right 
combination is to be found. Perhaps there is here 
a means of modifying what was paradoxical in the 
original hypothesis. 

Yet another observation. It never happens that 
unconscious work supplies ready-made the result of 
a lengthy calculation in which we have only to apply 
fixed rules. It might be supposed that the sub- 
liminal ego, purely automatic as it is, was peculiarly 
fitted for this kind of work, which is, in a sense, ex- 
clusively mechanical. It would seem that, by think- 
ing overnight of the factors of a multiplication sum, 
we might hope to find the product ready-made for 
us on waking ; or, again, that an algebraical calcula- 
tion, for instance, or a verification could be made 
unconsciously. Observation proves that such is by no 
means the case. All that we can hope from these 
inspirations, which are the fruits of unconscious 
work, is to obtain points of departure for such 
calculations. As for the calculations themselves, 
they must be made in the second period of conscious 
work which follows the inspiration, and in which 


the results of the inspiration are verified and the 
consequences deduced. The rules of these calcula- 
tions are strict and complicated ; they demand disci- 
pline, attention, will, and consequently consciousness. 
In the subliminal ego, on the contrary, there reigns 
what I would call liberty, if one could give this 
name to the mere absence of discipline and to dis- 
order born of chance. Only, this very disorder permits 
of unexpected couplings. 

I will make one last remark. When I related 
above some personal observations, I spoke of a night 
of excitement, on which I worked as though in spite 
of myself The cases of this are frequent, and it is 
not necessary that the abnormal cerebral activity 
should be caused by a physical stimulant, as in the 
case quoted. Well, it appears that, in these cases, 
we are ourselves assisting at our own unconscious 
work, which becomes partly perceptible to the over- 
excited consciousness, but does not on that account 
change its nature. We then become vaguely aware 
of what distinguishes the two mechanisms, or, if you 
will, of the methods of working of the two egos. 
The psychological observations I have thus suc- 
ceeded in making appear to me, in their general 
characteristics, to confirm the views I have been 

Truly there is great need of this, for in spite of 
everything they are and remain largely hypothetical. 
The interest of the question is so great that I do 
not regret having submitted them to the reader. 




" How can we venture to speak of the laws of chance ? 
Is not chance the antithesis of all law ? " It is thus 
that Bertrand expresses himself at the beginning of 
his "Calculus of Probabilities." Probability is the 
opposite of certainty ; it is thus what we are ignorant 
of, and consequently it would seem to be what we 
cannot calculate. There is here at least an apparent 
contradiction, and one on which much has already 
been written. 

To begin with, what is chance ? The ancients 
distinguished between the phenomena which seemed 
to obey harmonious laws, established once for all, 
and those that they attributed to chance, which were 
those that could not be predicted because they were 
not subject to any law. In each domain the precise 
laws did not decide everything, they only marked 
the limits within which chance was allowed to move. 
In this conception, the word chance had a precise, 
objective meaning ; what was chance for one was 
also chance for the other and even for the gods. 

But this conception is not ours. We have become 
complete determinists, and even those who wish to 


reserve the right of human free will at least allow 
determinism to reign undisputed in the inorganic 
world. Every phenomenon, however trifling it be, 
has a cause, and a mind infinitely powerful and 
infinitely well-informed concerning the laws of nature 
could have foreseen it from the beginning of the ages. 
If a being with such a mind existed, we could play 
no game of chance with him ; we should always 

For him, in fact, the word chance would have no 
meaning, or rather there would be no such thing as 
chance. That there is for us is only on account of 
our frailty and our ignorance. And even without 
going beyond our frail humanity, what is chance 
for the ignorant is no longer chance for the learned. 
Chance is only the measure of our ignorance. For- 
tuitous phenomena are, by definition, +hose whose 
laws we are ignorant of 

But is this definition very satisfactory? \Vhen the 
first Chaldean shepherds followed with their eyes 
the movements of the stars, they did not yet know 
the laws of astronomy, but would they have dreamed 
of saying that the stars move by chance? If a 
modern physicist is studying a new phenomenon, 
and if he discovers its law on Tuesday, would he 
have said on Monday that the phenomenon was 
fortuitous ? But more than this, do we not often 
invoke what Bertrand calls the laws of chance in 
order to predict a phenomenon ? For instance, in 
the kinetic theory of gases, we find the well-known 
laws of Mariotte and of Gay-Lussac, thanks to the 
hypothesis that the velocities of the gaseous mole- 
cules vary irregularly, that is to say, by chance. 

(1.777) 5 


The observable laws would be much less simple, 
say all the physicists, if the velocities were regulated 
by some simple elementary law, if the molecules 
were, as they say, organized, if they were subject to 
some discipline. It is thanks to chance — that is to 
say, thanks to our ignorance, that we can arrive at con- 
clusions. Then if the word chance is merely synony- 
mous with ignorance, vthat does this mean ? Must 
we translate as follows ? — 

"You ask me to predict the phenomena that will 
be produced. If I had the misfortune to know the 
laws of these phenomena, I could not succeed except 
by inextricable calculations, and I should have to 
give up the attempt to answer you ; but since I am 
fortunate enough to be ignorant of them, I will 
give you an answer at once. And, what is more 
extraordinary still, my answer will be right." 

Chance, then, must be something more than the 
name we' give to our ignorance. Among the phe- 
nomena whose causes we are ignorant of, we must 
distinguish between fortuitous phenomena, about 
which the calculation of probabilities will give us 
provisional information, and those that are not for- 
tuitous, about which we can say nothing, so long 
as we have not determined the laws that govern 
them. And as regards the fortuitous phenomena 
themselves, it is clear that the information that the 
calculation of probabilities supplies will not cease to 
be true when the phenomena are better known. 

The manager of a life insurance company does 
not know when each of the assured will die, but he 
relies upon the calculation of probabilities and on 
the law of large numbers, and he does not make a 


mistake, since he is able to pay dividends to his 
shareholders. These dividends would not vanish if 
a very far-sighted and very indiscreet doctor came, 
when once the policies were signed, and gave the 
manager information on the chances of life of the 
as.sured. The doctor would dissipate the ignorance 
of the manager, but he would have no effect upon 
the dividends, which are evidently not a result of 
that ignorance. 


In order to find the best definition of chance, we 
must examine some of the facts which it is agreed 
to regard as fortuitous, to which the calculation of 
probabilities seems to apply. We will then try to 
find their common characteristics. 

We will select unstable equilibrium as our first 
example. If a cone is balanced on its point, we know 
very well that it will fall, but we do not know to 
which side ; it seems that chance alone will decide. 
If the cone were perfectly symmetrical, if its axis 
were perfectly vertical, if it were subject to no other 
force but gravity, it would not fall at all. But the 
slightest defect of symmetry will make it lean slightly 
to one side or other, and as soon as it leans, be it 
ever so little, it will fall altogether to that side. 
Even if the symmetry is perfect, a very slight trepida- 
tion, or a breath of air, may make it incline a few 
seconds of arc, and that will be enough to determine 
its fall and even the direction of its fall, which will be 
that of the original inclination. 

A very small cause which escapes our notice 
determines a considerable effect that we cannot fail 
to see, and then we say that that effect is due to 


chance. If we knew exactly the laws of nature and 
the situation of the universe at the initial moment, 
we could predict exactly the situation of that same 
universe at a succeeding moment. But, even if it 
were the case that the natural laws had no longer 
any secret tor us, we could still only know the initial 
situation approximately. If that enabled us to predict 
the succeeding situation with the same approximation, 
that is all we require, and we should say that the 
phenomenon had been predicted, that it is governed 
by laws. But it is not always so ; it may happen that 
small differences in the initial conditions produce very 
great ones in the final phenomena. A small error in 
the former will produce an enormous error in the 
latter. Prediction becomes impossible, and we have 
the fortuitous phenomenon. 

Our second example will be very much like our 
first, and we will borrow it from meteorology. Why 
have meteorologists such difficulty in predicting the 
weather with any certainty ? Why is it that showers 
and even storms seem to come by chance, so that 
many people think it quite natural to pray for rain 
or fine weather, though they would consider it 
ridiculous to ask for an eclipse by prayer ? We see 
that great disturbances are generally produced in 
regions where the atmosphere is in unstable equilib- 
rium. The meteorologists see very well that the 
equilibrium is unstable, that a cyclone will be formed 
somewhere, but exactly where they are not in a 
position to say ; a tenth of a degree more or less at 
any given point, and the cyclone will burst here and 
not there, and extend its ravages over districts it 
would otherwise have spared. If they had been aware 


of this tenth of a degree, they could have known 
it beforehand, but the observations were neither 
sufficiently comprehensive nor sufficiently precise, and 
that is the reason why it all seems due to the 
intervention of chance. Here, again, we find the 
same contrast between a very trifling cause that 
is inappreciable to the observer, and considerable 
effects, that are sometimes terrible disasters. 

Let us pass to another example, the distribution of 
the minor planets on the Zodiac. Their initial 
longitudes may have had some definite order, but 
their mean motions were different and they have been 
revolving for so long that we may say that practically 
they are distributed by chance throughout the Zodiac. 
Very small initial differences in their distances from 
the sun, or, what amounts to the same thing, in their 
mean motions, have resulted in enormous differences 
in their actual longitudes. A difference of a thousandth 
part of a second in the mean daily motion will have 
the effect of a second in three years, a degree in ten 
thousand years, a whole circumference in three or 
four millions of years, and what is that beside the 
time that has elapsed since the minor planets became 
detached from Laplace's nebula ? Here, again, we 
have a small cause and a great effect, or better, small 
differences in the cause and great differences in the 

The eame of roulette does not take us so far as it 
might appear from the preceding example. Imagine 
a needle that can be turned about a pivot on a dial 
divided into a hundred alternate red and black 
sections. If the needle stops at a red section we win ; 
if not, we lose. Clearly, all depends on the initial 


impulse we give to the needle. I assume that the 
needle will make ten or twenty revolutions, but it 
will stop earlier or later according to the strength 
of the spin I have given it. Only a variation of a 
thousandth or a two-thousandth in the impulse is 
sufficient to determine whether my needle will stop 
at a black section or at the following section, which 
is red. These are differences that the muscular sense 
cannot appreciate, which would escape even more 
delicate instruments. It is, accordingly, impossible for 
me to predict what the needle I have just spun will 
do, and that is why my heart beats and I hope for 
everything from chance. The difference in the cause 
is imperceptible, and the difference in the effect is 
for me of the highest importance, since it affects my 
whole stake. 


In this connexion I wish to make a reflection that 
is somewhat foreign to my subject. Some years 
ago a certain philosopher said that the future was 
determined by the past, but not the past by the 
future ; or, in other words, that from the knowledge 
of the present we could deduce that of the future 
but not that of the past ; because, he said, one cause 
can produce only one effect, while tne same effect can 
be produced by several different causes. It is obvious 
that no scientist can accept this conclusion. The laws 
of nature link the antecedent to the consequent in 
such a way that the antecedent is determined by the 
consequent just as much as the consequent is by the 
antecedent. But what can have been the origin of 
the philosopher's error? We know that, in virtue 
of Carnot's principle, physical phenomena are irrevers- 


ible and that the world is tending towards uniformity. 
When two bodies of different temperatures are in 
conjunction, the warmer gives up heat to the colder, 
and accordingly we can predict that the temperatures 
will become equal. But once the temperatures have 
become equal, if we are asked about the previous state, 
what can we answer ? We can certainly say that one 
of the bodies was hot and the other cold, but we 
cannot guess which of the two was formerly the 

And yet in reality the temperatures never arrive 
at perfect equality. The difference between the 
temperatures only tends towards zero asymptotically. 
Accordingly there comes a moment when our 
thermometers are powerless to disclose it. But if 
we had thermometers a thousand or a hundred 
thousand times more sensitive, we should recognize 
that there is still a small difference, and that one of 
the bodies has remained a little warmer than the 
other, and then we should be able to state that this 
is the one which was formerly very much hotter than 
the other. 

So we have, then, the reverse of what we found in 
the preceding examples, great differences in the cause 
and small differences in the effect. Flammarion once 
imagined an observer moving away from the earth 
at a velocity greater than that of light. For him 
time would have its sign changed, history would be 
reversed, and Waterloo would come before Austerlitz. 
Well, for this observer effects and causes would be 
inverted, unstable equilibrium would no longer be the 
exception ; on account of the universal irreversibility, 
everything would seem to him to come out of a kind 


of chaos in unstable equilibrium, and the whole of 
nature would appear to him to be given up to chance. 


We come now to other arguments, in which we 
shall see somewhat different characteristics appearing, 
and first let us take the kinetic theory of gases. How 
are we to picture a receptacle full of gas ? Innumer- 
able molecules, animated with great velocities, course 
through the receptacle in all directions ; every moment 
they collide with the sides or else with one another, 
and these collisions take place under the most varied 
conditions. What strikes us most in this case is not 
the smallness of the causes, but their complexity. 
And yet the former element is still found here, and 
plays an important part. If a molecule deviated 
from its trajectory to left or right in a very small 
degree as compared with the radius of action of the 
gaseous molecules, it would avoid a collision, or would 
suffer it under different conditions, and that would 
alter the direction of its velocity after the collision 
perhaps by 90 or 180 degrees. 

That is not all. It is enough, as we have just seen, 
that the molecule should deviate before the collision 
in an infinitely small degree, to make it deviate after 
the collision in a finite degree. Then, if the molecule 
suffers two successive collisions, it is enough that it 
should deviate before the first collision in a degree of 
infinite smallness of the second order, to make it deviate 
after the first collision in a degree of infinite small- 
ness of the first order, and after the second collision 
in a finite degree. And the molecule will not suffer 
two collisions only, but a great number each second. 


So that if the first collision multiplied the deviation 
by a very large number, A, after n collisions it will be 
multiplied by A". It vi^ill, therefore, have become very 
great, not only because A is large — that is to say, 
because small causes produce great effects — but be- 
cause the exponent n is large, that is to say, because 
the collisions are very numerous and the causes very 

Let us pass to a second example. Why is it that 
in a shower the drops of rain appear to us to be 
distributed by chance ? It is again because of the 
complexity of the causes which determine their 
formation. Ions have been distributed through the 
atmosphere ; for a long time they have been sub- 
jected to constantly changing air currents, they have 
been involved in whirlwinds of very small dimensions, 
so that their final distribution has no longer any 
relation to their original distribution. Suddenly the 
temperature falls, the vapour condenses, and each of 
these ions becomes the centre of a raindrop. In 
order to know how these drops will be distributed 
and how many will fall on each stone of the pave- 
ment, it is not enough to know the original position 
of the ions, but we must calculate the effect of a 
thousand minute and capricious air currents. 

It is the same thing again if we take grains of dust 
in suspension in water. The vessel is permeated by 
currents whose law we know nothing of except that 
it is very complicated. After a certain length of 
time the grains will be distributed by chance, that 
is to say uniformly, throughout the vessel, and this 
is entirely due to the complication of the currents 
If they obeyed some simple law — if, for instance 


the vessel were revolving and the currents revolved 
in circles about its axis — the case would be altered, 
for each grain would retain its original height and 
its original distance from the axis. 

We should arrive at the same result by picturing 
the mixing of two liquids or of two fine powders. 
To take a rougher example, it is also what 
happens when a pack of cards is shuffled. At 
each shuffle the cards undergo a permutation similar 
to that studied in the theory of substitutions. 
What will be the resulting permutation? The prob- 
ability that it will be any particular permutation (for 
instance, that which brings the card occupying the 
position <^ {n) before the permutation into the position 
n), this probability, I say, depends on the habits of 
the player. But if the player shuffles the cards long 
enough, there will be a great number of successive 
permutations, and the final order which results will 
no longer be governed by anj'thing but chance ; I 
mean that all the possible orders will be equally 
probable. This result is due to the great number 
of successive permutations, that is to say, to the 
complexity of the phenomenon. 

A final word on the theory of errors. It is a case 
in which the causes have complexity and multiplicity. 
How numerous are the traps to which the observer 
is exposed, even with the best instrument. He must 
take pains to look out for and avoid the most flagrant, 
those which give birth to systematic errors. But 
when he has eliminated these, admitting that he 
succeeds in so doing, there still remain many which, 
though small, may become dangerous by the ac- 
cumulation of their effects. It is from these that 


accidental errors arise, and we attribute them to 
chance, because their causes are too complicated and 
too numerous. Here again we have only small causes, 
but each of them would only produce a small effect ; 
it is by their union and their number that their effects 
become formidable. 


There is yet a third point of view, which is less im- 
portant than the two former, on which I will not lay so 
much stress. When we are attempting to predict a 
fact and making an examination of the antecedents, 
we endeavour to enquire into the anterior situation. 
But we cannot do this for every part of the universe, 
and we are content with knowing what is going 
on in the neighbourhood of the place where the fact 
will occur, or what appears to have some connexion 
with the fact. Our enquiry cannot be complete, and 
we must know how to select. But we may happen 
to overlook circumstances which, at first sight, seemed 
completely foreign to the anticipated fact, to which 
we should never have dreamed of attributing any 
influence, which nevertheless, contrary to all anticipa- 
tion, come to play an important part. 

A man passes in the street on the way to his 
business. Some one familiar with his business could 
say what reason he had for starting at such an hour 
and why he went by such a street. On the roof a 
slater is at work. The contractor who employs him 
could, to a certain extent, predict what he will do. 
But the man has no thought for the slater, nor the 
slater for him ; they seem to belong to two worlds 
completely foreign to one another. Nevertheless 
the slater drops a tile v/hich kills the man, and we 


should have no hesitation in saying that this was 

Our frailty does not permit us to take in the whole 
universe, but forces us to cut it up in slices. We 
attempt to make this as little artificial as possible, 
and yet it happens, from time to time, that two of 
these slices react upon each other, and then the effects 
of this mutual action appear to us to be due to chance. 

Is this a third way of conceiving of chance ? Not 
always ; in fact, in the majority of cases, we come 
back to the first or second. Each time that two 
worlds, generally foreign to one another, thus come 
to act upon each other, the laws of this reaction 
cannot fail to be very complex, and moreover a very 
small change in the initial conditions of the two 
worlds would have been enough to prevent the 
reaction from taking place. How very little it would 
have taken to make the man pass a moment later, 
or the slater drop his tile a moment earlier ! 


Nothing that has been said so far explains why 
chance is obedient to laws. Is the fact that the 
causes are small, or that they are complex, sufficient 
to enable us to predict, if not what the effects will 
be m each case, at least what they will be on the 
average ? In order to answer this question, it will 
be best to return to some of the examples quoted 

I will begin with that of roulette. I said that the 
point where the needle stops will depend on the 
initial impulse given it. What is the probability that 
this impulse will be of any particular strength .? I 


do not know, but it is difficult not to admit that 
this probability is represented by a continuous 
analytical function. The probability that the impulse 
will be comprised between a and a + e will, then, 
clearly be equal to the probability that it will be 
comprised between a + e and a + ze, pi-ovided that € is 
very small. This is a property common to all 
analytical functions. Small variations of the function 
are proportional to small variations of the variable. 

But we have assumed that a very small variation in 
the impulse is sufficient to change the colour of the 
section opposite which the needle finally stops. 
From u to a + « is red, from a + e to a + 2e is black. 
The probability of each red section is accordingly the 
same as that of the succeeding black section, and 
consequently the total probability of red is equal 
to the total probability of black. 

The datum in the case is the analytical function 
which represents the probability of a particular 
initial impulse. But the theorem remains true, what- 
ever this datum may be, because it depends on a 
property common to all analytical functions. From 
this it results finally that we have no longer any need 
of the datum. 

What has just been said of the case of roulette 
applies also to the example of the minor planets. 
The Zodiac may be regarded as an immense roulette 
board on which the Creator has thrown a very great 
number of small balls, to which he has imparted 
different initial impulses, varying, however, according 
to some sort of law. Their actual distribution is 
uniform and independent of that law, for the same 
reason as in the preceding case. Thus we see why 


phenomena obey the laws of chance when small 
differences in the causes are sufficient to produce 
great differences in the effects. The probabilities of 
these small differences can then be regarded as 
proportional to the differences themselves, just be- 
cause these differences are small, and small increases 
of a continuous function are proportional to those 
of the variable. 

Let us pass to a totally different example, in which 
the complexity of the causes is the principal factor. 
I imagine a card-player shuffling a pack of cards. 
At each shuffle he changes the order of the cards, 
and he may change it in various ways. Let us take 
three cards only in order to simplify the explanation. 
The cards which, before the shuffle, occupied the 
positions 123 respectively may, after the shuffle, 
occupy the positions 

123, 231, 312, 321, 132, 213. 

Each of these six hypotheses is possible, and their 
probabilities are respectively 

/i. /a. /s, A. /s. A- 
The sum of these six numbers is equal to i, but that 
is all we know about them. The six probabilities 
natural!}' depend upon the player's habits, which we 
do not know. 

At the second shuffle the process is repeated, and 
under the same conditions. I mean, for instance, 
that p^ always represents the probability that the 
three cards which occupied the positions 123 after 
the n"' shuffle and before the w+i'", will occupy the 
positions 321 after the n+\"' shuffle. And this re- 
mains true, whatever the number n may be, since the 


player's habits and his method of shuffling remain 
the same. 

But if the number of shuffles is very large, the cards 
which occupied the positions 123 before the first shuffle 
may, after the last shuffle, occupy the positions 

123, 231, 312, 321, 132, 213, 
and the probability of each of these six hypotheses is 
clearly the same and equal to I- ; and this is true what- 
ever be the numbers A • • • A. which we do not know. 
The great number of shuffles, that is to say, the com- 
plexity of the causes, has produced uniformity. 

This would apply without change if there were more 
than three cards, but even with three the demonstra- 
tion would be complicated, so I will content myself 
with giving it for two cards only. We have now only 
two hypotheses 

12, 21, 

with the probabilities A and A = I -A- Assume that 
there are n shuffles, and that I win a shilling if the 
cards are finally in the initial order, and that I lose one 
if they are finally reversed. Then my mathematical 
expectation will be 

(A -A)" 

The difference A ~A is certainly smaller than i, so 
that if n is very large, the value of my expectation 
will be nothing, and we do not require to know A 
and A to know that the game is fair. 

Nevertheless there would be an exception if one of 
the numbers A and A was equal to i and the other to 
nothing. // would then hold good no longer, because 
our original hypotheses would be too simple. 

What we have just seen applies not only to the 


mixing of cards, but to all mixing, to that of powders 
and liquids, and even to that of the gaseous molecules 
in the kinetic theory of gases. To return to this theory, 
let us imagine for a moment a gas whose molecules 
cannot collide mutually, but can be deviated by col- 
lisions with the sides of the vessel in which the gas 
is enclosed. If the form of the vessel is sufficiently 
i.omplicated, it will not be long before the distribution 
of the molecules and that of their velocities become 
uniform. This will not happen if the vessel is spherical, 
or if it has the form of a rectangular parallelepiped. 
And why not? Because in the former case the dis- 
tance of any particular trajectory from the centre 
remains constant, and in the latter case we have 
the absolute value of the angle of each trajectory 
with the sides of the parallelepiped. 

Thus we see what we must understand by conditions 
that are too simple. They are conditions which pre- 
serve something of the original state as an invariable. 
Are the differential equations of the problem too 
simple to enable us to apply the laws of chance? 
This question appears at first sight devoid of any pre- 
cise meaning, but we know now what it means. They 
are too simple if something is preserved, if they 
admit a uniform integral. If something of the initial 
conditions remains unchanged, it is clear that the 
final situation can no longer be independent of the 
initial situation. 

We come, lastl}^ to the theory of errors. We are 
ignorant of what accidental errors are due to, and it is 
just because of this ignorance that we know they will 
obey Gauss's law. Such is the paradox. It is ex- 
plained in somewhat the same way as the preceding 


cases. We only need to know one thing — that the 
errors are very numerous, that they are very small, 
and that each of them can be equally well negative 
or positive. What is the curve of probability of each 
of them ? We do not know, but only assume that it 
is symmetrical. We can then show that the resultant 
error will follow Gauss's law, and this resultant law is 
independent of the particular laws which we do not 
know. Here again the simplicity of the result actually 
owes its existence to the complication of the data. 


But we have not come to the end of paradoxes. I 
recalled just above Flammarion's fiction of the man 
who travels faster than light, for whom time has its 
sign changed. I said that for him all phenomena 
would seem to be due to chance. This is true from 
a certain point of view, and yet, at any given moment, 
all these phenomena would not be distributed in con- 
formity with the laws of chance, since they would be 
just as they are for us, who, seeing them unfolded 
harmoniously and not emerging from a primitive 
chaos, do not look upon them as governed by chance. 

What does this mean ? For Flammarion's imagi- 
nary Lumen, small causes seem to produce great 
effects ; why, then, do things not happen as they do 
for us when we think we see great effects due to small 
causes ? Is not the same reasoning applicable to 
his case? 

Let us return to this reasoning. When small dif- 
ferences in the causes produce great differences in 
the effects, why are the effects distributed according 
to the laws of chance ? Suppose a difference of an 

(1,777) 6 


inch in the cause produces a difference of a mile in 
the effect. If I am to win in case the effect corre- 
sponds with a mile bearing an even number, my 
probability of winning will be -|. Why is this ? 
Because, in order that it should be so, the cause must 
correspond with an inch bearing an even number. 
Now, according to all appearance, the probability 
that the cause will vary between certain limits is 
proportional to the distance of those limits, provided 
that distance is very small. If this hypothesis be not 
admitted, there would no longer be any means of 
representing the probability by a continuous function. 
Now what will happen when great causes produce 
small effects ? This is the case in which we shall not 
attribute the phenomenon to chance, and in which 
Lumen, on the contrary, would attribute it to chance. 
A difference of a mile in the cause corresponds to 
a difference of an inch in the effect. Will the 
probability that the cause will be comprised between 
two limits n miles apart still be proportional to «? 
We have no reason to suppose it, since this dis- 
tance of n miles is great. But the probability that 
the effect will be comprised between two limits n 
inches apart will be precisely the same, and ac- 
cordingly it will not be proportional to n, and that 
notwithstanding the fact that this distance of n 
inches is small. There is, then, no means of repre- 
senting the law of probability of the effects by a 
continuous curve. I do not mean to say that the 
curve may not remain continuous in the mtalytical 
sense of the word. To infinitely small variations 
of the abscissa there will correspond infinitely small 
variations of the ordinate. But practically it would 


not be continuous, since to very small variations of 
the abscissa there would not correspond very small 
variations of the ordinate. It would become impos- 
sible to trace the curve with an ordinary pencil : that 
is what I mean. 

What conclusion are we then to draw ? Lumen has 
no right to say that the probability of the cause (that 
of his cause, which is our effect) must necessarily be 
represented by a continuous function. But if that be 
so, why have we the right ? It is because that state of 
unstable equilibrium that I spoke of just now as initial, 
is itself only the termination of a long anterior history. 
In the course of this history complex causes have been 
at work, and they have been at work for a long time. 
They have contributed to bring about the mixture oi" 
the elements, and they have tended to make everything 
uniform, at least in a small space. They have rounded 
off the corners, levelled the mountains, and filled up 
the valleys. However capricious and irregular the 
original curve they have been given, they have worked 
so much to regularize it that they will finally give us 
a continuous curve, and that is why we can quite con- 
fidently admit its continuity. 

Lumen would not have the same reasons for drawing 
this conclusion. For him complex causes would not 
appear as agents of regularity and of levelling ; on the 
contrary, they would only create differentiation and 
inequality. He would see a more and more varied 
world emerge from a sort of primitive chaos. The 
changes he would observe would be for him unfore- 
seen and impossible to foresee. They would seem 
to him due to some caprice, but that caprice would 
not be at all the same as our chance, since it would 


not be amenable to any law, while our chance has its 
own laws. All these points would require a much 
longer development, which would help us perhaps to 
a better comprehension of the irreversibility of the 


We have attempted to define chance, and it would 
be well now to ask ourselves a question. Has chance, 
thus defined so far as it can be, an objective character? 

We may well ask it. I have spoken of very small 
or very complex causes, but may not what is very 
small for one be great for another, and may not what 
seems very complex to one appear simple to another ? 
I have already given a partial answer, since I stated 
above most precisely the case in which differential 
equations become too simple for the laws of chance 
to remain applicable. But it would be well to exam- 
ine the thing somewhat more closely, for there are 
still other points of view we may take. 

What is the meaning of the word small ? To 
understand it, we have only to refer to what has 
been said above. A difference is very small, an 
interval is small, when within the limits of that in- 
terval the probability remains appreciably constant. 
Why can that probability be regarded as constant 
in a small interval? It is because we admit that the 
law of probability is represented by a continuous 
curve, not only continuous in the analytical sense of 
the word, but practically continuous, as I explained 
above. This means not only that it will present no 
absolute hiatus, but also that it will have no projections 
or depressions too acute or too much accentuated. 

What gives us the right to make this hypothesis? 


As I said above, it is because, from the begihning of 
the ages, there are complex causes that never cease 
to operate in the same direction, which cause the 
world to tend constantly towards uniformity without 
the possibility of ever going back. It is these causes 
which, little by little, have levelled the projections and 
filled up the depressions, and it is for this reason that 
our curves of probability present none but gentle undu- 
lations. In millions and millions of centuries we shall 
have progressed another step towards uniformity, and 
these undulations will be ten times more gentle still. 
The radius of mean curvature of our curve will have 
become ten times longer. And then a length that 
to-day does not seem to us very small, because an 
arc of such a length cannot be regarded as rectilineal, 
will at that period be properly qualified as very small, 
since the curvature will have become ten times less, 
and an arc of such a length will not differ appreciably 
from a straight line. 

Thus the word very small remains relative, but it 
is not relative to this man or that, it is relative to 
the actual state of the world. It will change its 
meaning when the world becomes more uniform and 
all things are still more mixed. But then, no doubt, 
men will no longer be able to live, but will have to 
make way for other beings, shall I say much smaller 
or much larger? So that our criterion, remaining 
true for all men, retains an objective meaning. 

And, further, what is the meaning of the word very 
complex ? I have already given one solution, that 
which I referred to again at the beginning of this 
section ; but there are others. Complex causes, I have 
said, produce a more and more intimate mixture, but 


how long will it be before this mixture satisfies us ? 
When shall we have accumulated enough complica- 
tions ? When will the cards be sufficiently shuffled ? 
If we mix two powders, one blue and the other white, 
there comes a time when the colour of the mixture 
appears uniform. This is on account of the infirmity 
of our senses ; it would be uniform for the long- 
sighted, obliged to look at it from a distance, when 
it would not yet be so for the short-sighted. Even 
when it had become uniform for all sights, we could 
still set back the limit by employing instruments. 
There is no possibility that any man will ever dis- 
tinguish the infinite variety that is hidden under the 
uniform appearance of a gas, if the kinetic theory is 
true. Nevertheless, if we adopt Gouy's ideas on the 
Brownian movement, does not the microscope seem to 
be on the point of showing us something analogous ? 

This new criterion is thus relative like the first, and 
if it preserves an objective character, it is because all 
men have about the same senses, the power of their 
instruments is limited, and, moreover, they only make 
use of them occasionally. 


It is the same in the moral sciences, and particularly 
in history. The historian is obliged to make a selec- 
tion of the events in the period he is studying, and he 
only recounts those that seem to him the most im- 
portant. Thus he contents himself with relating the 
most considerable events of the i6th century, for 
instance, and similarly the most remarkable facts of 
the 17th century. If the former are sufficient to 
explain the latter, we say that these latter conform 


to the laws of history. But if a great event of the 
17th century owes its cause to a small fact of the 
l6th century that no history reports and that every 
one has neglected, then we say that this event is due 
to chance, and so the word has the same sense as in 
the physical sciences ; it means that small causes 
have produced great effects. 

The greatest chance is the birth of a great man. 
It is only by chance that the meeting occurs of two 
genital cells of different sex that contain precisely, 
each on its side, the mysterious elements whose mutual 
reaction is destined to produce genius. It will be 
readily admitted that these elements must be rare^ 
and that their meeting is still rarer. How little it 
would have taken to make the spermatozoid which 
carried them deviate from its course. It would have 
been enough to deflect it a hundredth part of a inch, 
and Napoleon would not have been born and the 
destinies of a continent would have been changed. 
No example can give a better comprehension of the 
true character of chance. 

One word more about the paradoxes to which the 
application of the calculation of probabilities to the 
moral sciences has given rise. It has been demon- 
strated that no parliament would ever contain a 
single member of the opposition, or at least that such 
an event would be so improbable that it would be 
quite safe to bet against it, and to bet a million to 
one. Condorcet attempted to calculate how many 
jurymen it would require to make a miscarriage of 
justice practically impossible. If we used the results 
of this calculation, we should certainly be exposed 
to the same disillusionment as by betting on the 


strength of the calculation that the opposition would 
never have a single representative. 

The laws of chance do not apply to these questions. 
If justice does not always decide on good grounds, 
it does not make so much use as is generally supposed 
of Bridoye's method. This is perhaps unfortunate, 
since, if it did, Condorcet's method would protect us 
against miscarriages. 

What does this mean ? We are tempted to attribute 
facts of this nature to chance because their causes 
are obscure, but this is not true chance. The causes 
are unknown to us, it is true, and they are even 
complex ; but they are not sufficiently complex, since 
they preserve something, and we have seen that this 
is the distinguishing mark of "too simple" causes. 
When men are brought together, they no longer 
decide by chance and independently of each other, 
but react upon one another. Many causes come into 
action, they trouble the men and draw them this way 
and that, but there is one thing they cannot destroy, 
the habits they have of Panurge's sheep. And it is this 
that is preserved. 


The application of the calculation of probabilities 
to the exact sciences also involves many difficulties. 
Why are the decimals of a table of logarithms or of 
the number x distributed in accordance with the laws 
of chance ? I have elsewhere studied the question 
in regard to logarithms, and there it is eas}'. It is 
clear that a small difference in the argument will give 
a small difference in the logarithm, but a great differ- 
ence in the sixth decimal of the logarithm. We still 
find the same criterion. 


But as regards the number tt the question presents 
more difficulties, and for the moment I have no 
satisfactory explanation to give. 

There are many other questions that might be 
raised, if I wished to attack them before answering 
the one I have more especially set myself When we 
arrive at a simple result, when, for instance, we find 
a round number, we say that such a result cannot be 
due to chance, and we seek for a non-fortuitous cause 
to explain it. And in fact there is only a very slight 
likelihood that, out of 10,000 numbers, chance will 
give us a round number, the number 10,000 for in- 
stance ; there is only one chance in 10,000. But 
neither is there more than one chance in 10,000 that 
it will give us any other particular number, and yet 
this result does not astonish us, and we feel no hesita- 
tion about attributing it to chance, and that merely 
because it is less striking. 

Is this a simple illusion on our part, or are there 
cases in which this view is legitimate ? We must 
hope so, for otherwise all science would be impossible. 
When we wish to check a hypothesis, what do we 
do? We cannot verify all its consequences, since 
they are infinite in number. We content ourselves 
with verifying a few, and, if we succeed, we declare 
that the hypothesis is confirmed, for so much success 
could not be due to chance. It is always at bottom 
the same reasoning. 

I cannot justify it here completely, it would take 
me too long, but I can say at least this. We find 
ourselves faced by two hypotheses, either a simple 
cause or else that assemblage of complex causes we 
call chance. We find it natural to admit that the 


former must produce a simple result, and then, if we 
arrive at this simple result, the round number for 
instance, it appears to us more reasonable to attribute 
it to the simple cause, which was almost certain to 
give it us, than to chance, which could only give it 
us once in 10,000 times. It will not be the same 
if we arrive at a result that is not simple. It is true 
that chance also will not give it more than once in 
10,000 times, but the simple cause has no greater 
chance of producing it. 





It is impossible to picture empty space. All our 
efforts to imagine pure space from which the changing 
images of material objects are excluded can only 
result in a representation in which highly-coloured 
surfaces, for instance, are replaced by lines of slight 
colouration, and if we continued in this direction to the 
end, everything would disappear and end in nothing. 
-Hence arises the irreducible relativity of space. 

Whoever speaks of absolute space uses a word de- 
void of meaning. This is a truth that has been long 
proclaimed by all who have reflected on the question, 
but one which we are too often inclined to forget. 

If I am at a definite point in Paris, at the Place 
du Pantheon, for instance, and I say, " I will come 
back here to-morrow ; " if I am asked, " Do you mean 
that you will come back to the same point in space?" 
I should be tempted to answer yes. Yet I should 
be wrong, since between now and to-morrow the earth 
will have moved, carrying with it the Place du Pan- 
theon, which will have travelled more than a million 
miles. And if I wished to speak more accurately, I 
should gain nothing, since this million of miles has 


been covered by our globe in its motion in relation 
to the sun, and the sun in its turn moves in relation 
to the Milky Way, and the Milky Way itself is no 
doubt in motion without our being able to recognize 
its velocity. So that we are, and shall always be, 
completely ignorant how far the Place du Pantheon 
moves in a day. In fact, what I meant to say was, 
" To-morrow I shall see once more the dome and 
pediment of the Pantheon," and if there was no 
Pantheon my sentence would have no meaning" and 
space would disappear. 

This is one of the most commonplace forms of the 
principle of the relativity of space, but there is another 
on which Delbeuf has laid particular stress. Suppose 
that in one night all the dimensions of the universe 
became a thousand times larger. The world will* 
remain similar to itself, if we give the word similitude 
the meaning it has in the third book of Euclid. 
Only, what was formerly a metre long will now measure 
a kilometre, and what was a millimetre long will 
become a metre. The bed in which I went to sleep 
and my body itself will have grown in the same 
proportion. When I wake in the morning what will 
be my feeling in face of such an astonishing trans- 
formation ? Well, I shall not notice anything at all. 
The most exact measures will be incapable of revealing 
anything of this tremendous change, since the yard- 
measures I shall use will have varied in exactly the 
same proportions as the objects I shall attempt to 
measure. In reality the change only exists for those 
who argue as if space were absolute. If I have argued 
for a moment as they do, it was only in order to make 
it clearer that their view implies a contradiction. In 


reality it would be better to say that as space is 
relative, nothing at all has happened, and that it is 
for that reason that we have noticed nothing. 

Have we any right, therefore, to say that we know 
the distance between two points? No, since that 
distance could undergo enormous variations without 
our being able to perceive it, provided other distances 
varied in the same proportions. We saw just now 
that when I say I shall be here to-morrow, that does 
not mean that to-morrow I shall be at the point in 
space where I am to-day, but that to-morrow I shall 
be at the same distance from the Panthdon as I am 
to-day. And already this statement is not sufficient, 
and I ought to say that to-morrow and to-day my 
distance from the Pantheon will be equal to the same 
number of times the length of my body. 

But that is not all. I imagined the dimensions of 
the world changing, but at least the world remaining 
always similar to itself We can go much further than 
that, and one of the most surprising theories of modern 
physicists will furnish the occasion. According to 
a hypothesis of Lorentz and Fitzgerald,* all bodies 
carried forward in the earth's motion undergo a de- 
formation. This deformation is, in truth, very slight, 
since all dimensions parallel with the earth's motion 
are diminished by a hundred-millionth, while dimen- 
sions perpendicular to this motion are not altered. 
But it matters little that it is slight ; it is enough 
that it should exist for the conclusion I am soon 
going to draw from it. Besides, though I said that 
it is slight, I really know nothing about it. I have 
myself fallen a victim to the tenacious illusion that 

* Vide infra. Book III. Chap. ii. 


makes us believe that we think of an absolute space. 
I was thinking of the earth's motion on its elliptical 
orbit round the sun, and I allowed 1 8 miles a second 
for its velocity. But its true velocity (I mean this 
time, not its absolute velocity, which has no sense, 
but its velocity in relation to the ether), this I do not 
know and have no means of knowing. It is, perhaps, 
lo or lOO times as high, and then the deformation 
will be 100 or io,000 times as great. 

It is evident that we cannot demonstrate this de- 
formation. Take a cube with sides a yard long. It 
is deformed on account of the earth's velocity ; one 
of its sides, that parallel with the motion, becomes 
smaller, the others do not vary. If I wish to assure 
myself of this with the help of a yard-measure, I shall 
measure first one of the sides perpendicular to the 
motion, and satisfy myself that my measure fits this 
side exactly ; and indeed neither one nor other of 
these lengths is altered, since they are both perpendic- 
ular to the motion. I then wish to measure the other 
side, that parallel with the motion ; for this purpose 
I change the position of my measure, and turn it so 
as to apply it to this side. But the yard-measure, 
having changed its direction and having become paral- 
lel with the motion, has in its turn undergone the 
deformation, so that, though the side is no longer a 
yard long, it will still fit it exactly, and I shall be 
aware of nothing. 

What, then, I shall be asked, is the use of the 
hypothesis of Lorentz and Fitzgerald if no experiment 
can enable us to verify it ? The fact is that my state- 
ment has been incomplete. I have only spoken of 
measurements that can be made with a yard-measure. 


but we can also measure a distance by the time that 
light takes to traverse it, on condition that we admit 
that the velocity of light is constant, and independent 
of its direction. Lorentz could have accounted for the 
facts by supposing that the velocity of light is greater 
in the direction of the earth's motion than in the 
perpendicular direction. He preferred to admit that 
the velocity is the same in the two directions, but that 
bodies are smaller in the former than in the latter. If 
the surfaces of the waves of light had undergone the 
same deformations as material bodies, we should never 
have perceived the Lorentz-Fitzgerald deformation. 

In the one case as in the other, there can be no 
question of absolute magnitude, but of the meas- 
urement of that magnitude by means of some instru- 
ment. This instrument may be a yard-measure or 
the path traversed by light It is only the relation 
of the magnitude to the instrument that we measure, 
and if this relation is altered, we have no means of 
knowing whether it is the magnitude or the instrument 
that has changed. 

But what I wish to make clear is, that in this 
deformation the world has not remained similar to 
itself. Squares have become rectangles or parallel- 
ograms, circles ellipses, and spheres ellipsoids. And 
yet we have no means of knowing whether this de- 
formation is real. 

It is clear that we might go much further. Instead 
of the Lorentz-Fitzgerald deformation, with its ex- 
tremely simple laws, we might imagine a deformation 
of any kind whatever ; bodies might be deformed in 
accordance with any laws, as complicated as we liked, 
and we should not perceive it, provided all bodies 

(1,777) 7 


without exception were deformed in accordance with 
the same laws. When I say all bodies without excep- 
tion, I include, of course, our own bodies and the rays 
of light emanating from the different objects. 

If we look at the world in one of those mirrors 
of complicated form which deform objects in an odd 
way, the mutual relations of the different parts of the 
world are not altered ; if, in fact, two real objects 
touch, their images likewise appear to touch. In truth, 
when we look in such a mirror we readily perceive the 
deformation, but it is because the real world exists 
beside its deformed image. And even if this real 
world were hidden from us, there is something which 
cannot be hidden, and that is ourselves. We cannot 
help seeing, or at least feeling, our body and our 
members which have not been deformed, and continue 
to act as measuring instruments. But if we imagine 
our body itself deformed, and in the same way as if 
it were seen in the mirror, these measuring instruments 
will fail us in their turn, and the deformation will 
no longer be able to be ascertained. 

Imagine, in the same way, two universes which are 
the image one of the other. With each object P in 
the universe A, there corresponds, in the universe B, 
an object P^ which is its image. The co-ordinates 
of this image P^ are determinate functions of those 
of the object P ; moreover, these functions may be 
of any kind whatever — I assume only that they are 
chosen once for all. Between the position of P and 
that of P^ there is a constant relation ; it matters little 
what that relation may be, it is enough that it should 
be constant. 

Well, these two universes will be indistinguishable. 



I mean to say that the former will be for its inhab- 
itants what the second is for its own. This would 
be true so long as the two universes remained foreign 
to one another. Suppose we are inhabitants of the 
universe A ; we have constructed our science and 
particularly our geometry. During this time the in- 
habitants of the universe B have constructed a science, 
and as their world is the image of ours, their geometry 
will also be the image of ours, or, more accurately, 
it will be the same. But if one day a window were to 
open for us upon the universe B, we should feel 
contempt for them, and we should say, "These 
wretched people imagine that they have made a 
geometry, but what they so name is only a grotesque 
image of ours ; their straight lines are all twisted, 
their circles are hunchbacked, and their spheres have 
capricious inequalities." We should have no suspicion 
that they were saying the same of us, and that no 
one will ever know which is right. 

We see in how large a sense we must understand 
the relativity of space. Space is in reality amorphous, 
and it is only the things that are in it that give it 
a form. What are we to think, then, of that direct 
intuition we have of a straight line or of distance ? 
We have so little the intuition of distance in itself 
that, in a single night, as we have said, a distance 
could become a thousand times greater without our 
being able to perceive it, if all other distances had 
undergone the same alteration. And in a night the 
universe B might even be substituted for the universe 
A without our having any means of knowing it, and 
then the straight lines of yesterday would have ceased 
to be straight, and we should not be aware of anything. 


One part of space is not by itself and in the absolute 
sense of the word equal to another part of space, for 
if it is so for us, it will not be so for the inhabitants of 
the universe B, and they have precisely as much right 
to reject our opinion as we have to condemn theirs. 

I have shown elsewhere what are the consequences 
of these facts from the point of view of the idea that 
we should construct non-Euclidian and other analogous 
geometries. I do not wish to return to this, and 
I will take a somewhat different point of view. 


If this intuition of distance, of direction, of the 
straight line, if, in a word, this direct intuition of space 
does not exist, whence comes it that we imagine 
we have it? If this is only an illusion, whence comes 
it that the illusion is so tenacious ? This is what 
we must examine. There is no direct intuition of 
magnitude, as we have said, and we can only arrive 
at the relation of the magnitude to our measuring 
instruments. Accordingly we could not have con- 
structed space if we had not had an instrument 
for measuring it. Well, that instrument to which we 
refer everything, which we use instinctively, is our 
own body. It is in reference to our own body that we 
locate exterior objects, and the only special relations 
of these objects that we can picture to ourselves are 
their relations with our body. It is our body that 
serves us, so to speak, as a system of axes of 

For instance, at a moment a the presence of an 
object A is revealed to me by the sense of sight ; at 
another moment /i the presence of another object 


B is revealed by another sense, that, for instance, 
of hearing or of touch. I judge that this object B 
occupies the same place as the object A. What does 
this mean? To begin with, it does not imply that 
these two objects occupy, at two different moments, 
the same point in an absolute space, which, even 
if it existed, would escape our knowledge, since 
between the moments a and fi the solar system has 
been displaced and we cannot know what this dis- 
placement is. It means that these two objects occupy 
the same relative position in reference to our body. 

But what is meant even by this ? The impressions 
that have come to us from these objects have followed 
absolutely different paths — the optic nerve for the 
object A, and the acoustic nerve for the object B ; 
they have nothing in common from the qualitative 
point of view. The representations we can form of 
these two objects are absolutely heterogeneous and 
irreducible one to the other. Only I know that, 
in order to reach the object A, I have only to extend 
my right arm in a certain way ; even though I refrain 
from doing it, I represent to myself the muscular and 
other analogous sensations which accompany that 
extension, and that representation is associated with 
that of the object A. 

Now I know equally that I can reach the object B 
by extending my right arm in the same way, an 
extension accompanied by the same train of muscular 
sensations. And I mean nothing else but this when 
I say that these two objects occupy the same 

I know also that I could have reached the object A 
by another appropriate movement of the left arm, 


and I represent to myself the muscular sensations that 
would have accompanied the movement. And by 
the same movement of the left arm, accompanied 
by the same sensations, I could equally have reached 
the object B. 

And this is very important, since it is in this 
way that I could defend myself against the dangers 
with which the object A or the object B might threaten 
me. With each of the blows that may strike us, 
nature has associated one or several parries which 
enable us to protect ourselves against them. The 
same parry may answer to several blows. It is 
thus, for instance, that the same movement of the 
right arm would have enabled us to defend our- 
selves at the moment a against the object A, and 
at the moment /3 against the object B. Similarly, the 
same blow may be parried in several ways, and we 
have said, for instance, that we could reach the object 
A equally well either by a certain movement of the 
right arm, or by a certain movement of the left. 

All these parries have nothing in common with one 
another, except that they enable us to avoid the same 
blow, and it is that, and nothing but that, we 
mean when we say that they are movements ending 
in the same point in space. Similarly, these objects, 
of which we say that they occupy the same point in 
space, have nothing in common, except that the same 
parry can enable us to defend ourselves against them. 

Or, if we prefer it, let us imagine innumerable 
telegraph wires, some centripetal and others centri- 
fugal. The centripetal wires warn us of accidents 
that occur outside, the centrifugal wires have to 
provide the remedy. Connexions are established 


in such a way that when one of the centripetal wires 
is traversed by a current, this current acts on a central 
exchange, and so excites a current in one of the 
centrifugal wires, and matters are so arranged that 
several centripetal wires can act on the same centri- 
fugal wire, if the same remedy is applicable to several 
evils, and that one centripetal wire can disturb several 
centrifugal wires, either simultaneously or one in 
default of the other, every time that the same evil 
can be cured by several remedies. 

It is this complex system of associations, it is this 
distribution board, so to speak, that is our whole 
geometry, or, if you will, all that is distinctive in our 
geometry. What we call our intuition of a straight 
line or of distance is the consciousness we have of 
these associations and of their imperious character. 

Whence this imperious character itself comes, it 
is easy to understand. The older an association is, 
the more indestructible it will appear to us. But 
these associations are not, for the most part, conquests 
made by the individual, since we see traces of them 
in the newly-born infant; they are conquests made 
by the race. The more necessary these conquests 
were, the more quickly they must have been brought 
about by natural selection. 

On this account those we have been speaking 
of must have been among the earliest, since without 
them the defence of the organism would have been 
impossible. As soon as the cells were no longer 
merely in juxtaposition, as soon as they were called 
upon to give mutual assistance to each other, some 
such mechanism as we have been describing must 
necessarily have been organized in order that the 


assistance should meet the danger without mis- 

When a frog's head has been cut off, and a drop of 
acid is placed at some point on its skin, it tries 
to rub off the acid with the nearest foot ; and if that 
foot is cut off, it removes it with the other foot. Here 
we have, clearly, that double parry I spoke of just now, 
making it possible to oppose an evil by a second 
remedy if the first fails. It is this multiplicity of 
parries, and the resulting co-ordination, that is space. 

We see to what depths of unconsciousness we have 
to descend to find the first traces of these spacial 
associations, since the lowest parts of the nervous 
system alone come into play. Once we have rea- 
lized this, how can we be astonished at the resistance 
we oppose to any attempt to dissociate what has been 
so long associated ? Now, it is this very resistance 
that we call the evidence of the truths of geometry. 
This evidence is nothing else than the repugnance we 
feel at breaking with very old habits with which we 
have always got on very well. 


The space thus created is only a small space that 
does not extend beyond what my arm can reach, 
and the intervention of memory is necessary to set 
back its limits. There are points that will always 
remain out of my reach, whatever effort I may make 
to stretch out my hand to them. If I were attached 
to the ground, like a sea-polype, for instance, which 
can only extend its tentacles, all these points would 
be outside space, since the sensations we might 
experience from the action of bodies placed there 


would not be associated with the idea of any move- 
ment enabling us to reach them, or with any appro- 
priate parry. These sensations would not seem to us 
to have any spacial character, and we should not 
attempt to locate them. 

But we are not fixed to the ground like the inferior 
animals. If the enemy is too far off, we can advance 
upon him first and extend our hand when we are near 
enough. This is still a parry, but a long-distance 
parry. Moreover, it is a complex parry, and into the 
representation we make of it there enter the repre- 
sentation of the muscular sensations caused by the 
movement of the legs, that of the muscular sensations 
caused by the final movement of the arm, that of the 
sensations of the semi-circular canals, etc. Besides, we 
have to make a representation, not of a complexus 
of simultaneous sensations, but of a complexus of 
successive sensations, following one another in a deter- 
mined order, and it is for this reason that I said just 
now that the intervention of memory is necessary. 

We must further observe that, to reach the same 
point, I can approach nearer the object to be attained, 
in order not to have to extend my hand so far. And 
how much more might be said ? It is not one only, but 
a thousand parries I can oppose to the same danger. 
All these parries are formed of sensations that may 
have nothing in common, and yet we regard them 
as defining the same point in space, because they can 
answer to the same danger and are one and all 
of them associated with the notion of that danger. It 
is the possibility of parrying the same blow which 
makes the unity of these different parries, just as 
it is the possibility of being parried in the same way 


which makes the unity of the blows of such different 
kinds that can threaten us from the same point 
in space. It is this double unity that makes the 
individuality of each point in space, and in the notion 
of such a point there is nothing else but this. 

The space I pictured in the preceding section, 
which I might call restricted space, was referred to 
axes of co-ordinates attached to my body. These axes 
were fixed, since my body did not move, and it 
was only my limbs that changed their position. What 
are the axes to which the extended space is naturally 
referred — that is to say, the new space I have just 
defined ? We define a point by the succession of 
movements we require to make to reach it, starting 
from a certain initial position of the body. The axes 
are accordingly attached to this initial position of the 

But the position I call initial may be arbitrarily 
chosen from among all the positions my body has 
successively occupied. If a more or less unconscious 
memory of these successive positions is necessary for 
the genesis of the notion of space, this memory can go 
back more or less into the past. Hence results a 
certain indeterminateness in the very definition of 
space, and it is precisely this indeterminateness which 
constitutes its relativity. 

Absolute space exists no longer ; there is only space 
relative to a certain initial position of the body. For 
a conscious being, fixed to the ground like the inferior 
animals, who would consequently only know restricted 
space, space would still be relative, since it would be 
referred to his body, but this being would not be 
conscious of the relativity, because the axes to which 


he referred this restricted space would not change. 
No doubt the rock to which he was chained would 
not be motionless, since it would be involved in the 
motion of our planet ; for us, consequently, these axes 
would change every moment, but for him they would 
not change. We have the faculty of referring our 
extended space at one time to the position A of our 
body considered as initial, at another to the position 
B which it occupied some moments later, which we 
are free to consider in its turn as initial, and, accord- 
ingly, we make unconscious changes in the co-ordinates 
every moment. This faculty would fail our imaginary 
being, and, through not having travelled, he would 
think space absolute. Every moment his system of 
axes would be imposed on him ; this system might 
change to any extent in reality, for him it would be 
always the same, since it would always be the unique 
system. It is not the same for us who possess, each 
moment, several systems between which we can choose 
at will, and on condition of going back by memory 
more or less into the past. 

That is not all, for the restricted space would not 
be homogeneous. The different points of this space 
could not be regarded as equivalent, since some could 
only be reached at the cost of the greatest efforts, 
while others could be reached with ease. On the 
contrary, our extended space appears to us homoge- 
neous, and we say that all its points are equivalent. 
What does this mean ? 

If we start from a certain position A, we can, 
starting from that position, effect certain movements 
M, characterized by a certain complexus of muscular 
sensations. But, starting from another position B, 


we can execute movements M^ which will be char- 
acterized by the same muscular sensations. Then let 
a be the situation of a certain point in the body, the 
tip of the forefinger of the right hand, for instance, 
in the initial position A, and let b be the position of 
this same forefinger when, starting from that position 

A, we have executed the movements M. Then let d^ 
be the situation of the forefinger in the position B, 
and b^ its situation when, starting from the position 

B, we have executed the movements M^ 

Well, I am in the habit of saying that the points a 
and b are, in relation to each other, as the points a)- 
and b^, and that means simply that the two series of 
movements M and M^ are accompanied by the same 
muscular sensations. And as I am conscious that, 
in passing from the position A to the position B, my 
body has remained capable of the same movements, 
I know that there is a point in space which is to the 
point a'- what some point b is to the point a, so that 
the two points a and a)- are equivalent. It is this that 
is called the homogeneity of space, and at the same 
time it is for this reason that space is relative, since 
its properties remain the same whether they are 
referred to the axes A or to the axes B. So that the 
relativity of space and its homogeneity are one and 
the same thing. 

Now, if I wish to pass to the great space, which is 
no longer to serve for my individual use only, but in 
which I can lodge the universe, I shall arrive at it by 
an act of imagination. I shall imagine what a giant 
would experience who could reach the planets in a 
few steps, or, if we prefer, what I should feel myself 
in presence of a world in miniature, in which these 


planets would be replaced by little balls, while on 
one of these little balls there would move a Lilliputian 
that I should call myself. But this act of imagination 
would be impossible for me if I had not previously 
constructed my restricted space and my extended 
space for my personal use. 


Now we come to the question why all these spaces 
have three dimensions. Let us refer to the " distribu- 
tion board" spoken of above. We have, on the one 
side, a list of the different possible dangers — let us 
designate them as Ai, A 2, etc. — and, on the other side, 
the list of the different remedies, which I will call in 
the same way Bi, B2, etc. Then we have connexions 
between the contact studs of the first list and those of 
the second in such a way that when, for instance, the 
alarm for danger A3 works, it sets in motion or 
may set in motion the relay corresponding to the 
parry B4. 

As I spoke above of centripetal or centrifugal wires, 
I am afraid that all I have said may be taken, not as 
a simple comparison, but as a description of the 
nervous system. Such is not my thought, and that 
for several reasons. Firstly, I should not presume to 
pronounce an opinion on the structure of the nervous 
system which I do not know, while those who have 
studied it only do so with circumspection. Secondly, 
because, in spite of my incompetence, I fully realize 
that this scheme would be far too simple. And lastly, 
because, on my list of parries, there appear some that 
are very complex, which may even, in the case of 
extended space, as we have seen above, consist of 


several steps followed by a movement of the arm. It 
is not a question, then, of physical connexion between 
two real conductors, but of psychological association 
between two series of sensations. 

If A I and A2, for instance, are both of them 
associated with the parry Bl, and if Al is similarly 
associated with B2, it will generally be the case that 
A2 and B2 will also be associated. If this fundamental 
law were not generally true, there would only be an 
immense confusion, and there would be nothing that 
could bear any resemblance to a conception of space 
or to a geometry. How, indeed, have we defined a 
point in space ? We defined it in two ways : on the 
one hand, it is the whole of the alarms A which are 
in connexion with the same parry B ; on the other, 
it is the whole of the parries B which are in connexion 
with the same alarm A. If our law were not true, we 
should be obliged to say that A I and A2 correspond 
with the same point, since they are both in con- 
nexion with Bl ; but we should be equally obliged 
to say that they do not correspond with the same 
point, since Ai would be in connexion with B2, and 
this would not be true of A2 — which would be a 

But from another aspect, if the law were rigorously 
and invariably true, space would be quite different 
from what it is. We should have well-defined cate- 
gories, among which would be apportioned the alarms 
A on the one side and the parries B on the other. 
These categories would be exceedingly numerous, but 
they would be entirely separated one from the other. 
Space would be formed of points, very numerous but 
discrete ; it would be discontinuous. There would be 


no reason for arranging these points in one order 
rather than another, nor, consequently, for attributing 
three dimensions to space. 

But this is not the case. May I be permitted for 
a moment to use the language of those who know 
geometry already? It is necessary that I should do 
so, since it is the language best understood by those 
to whom I wish to make myself clear. When I wish 
to parry the blow, I try to reach the point whence 
the blow comes, but it is enough if I come fairly near 
it. Then the parry Bi may answer to Ai, and to 
A2 if the point which corresponds with Bi is sufficiently 
close both to that which corresponds with Ai and to 
that which corresponds with A2. But it may happen 
that the point which corresponds with another parry 
B2 is near enough to the point corresponding with 
A I, and not near enough to the point corresponding 
with A2. And so the parry B2 may answer to Ai 
and not be able to answer to A 2. 

For those who do not yet know geometry, this may 
be translated simply by a modification of the law 
enunciated above. Then what happens is as follows. 
Two parries, Bi and B2, are associated with one alarm 
A I, and with a very great number of alarms that we 
will place in the same category as A I, and make to 
correspond with the same point in space. But we 
may find alarms A2 which are associated with B2 and 
not with Bi, but on the other hand are associated with 
B3, which are not with Ai, and so on in succession, 
so that we may write the sequence 

Bi, Ai, B2, A2, B3, A3, B4, A4, 
in which each term is associated with the succeeding 


and preceding terms, but not with those that are 
several places removed. 

It is unnecessary to add that each of the terms of 
these sequences is not isolated, but forms part of a 
very numerous category of other alarms or other 
parries which has the same connexions as it, and 
may be regarded as belonging to the same point in 
space. Thus the fundamental law, though admitting 
of exceptions, remains almost always true. Only, in 
consequence of these exceptions, these categories, 
instead of being entirely separate, partially encroach 
upon each other and mutually overlap to a certain 
extent, so that space becomes continuous. 

Furthermore, the order in which these categories 
must be arranged is no longer arbitrary, and a 
reference to the preceding sequence will make it 
clear that B2 must be placed between Ai and A2, 
and, consequently, between Br and B3, and that it 
could not be placed, for instance, between B3 
and B4. 

Accordingly there is an order in which our cate- 
gories range themselves naturally which corresponds 
with the points in space, and experience teaches us 
that this order presents itself in the form of a three- 
circuit distribution board, and it is for this reason 
that space has three dimensions. 


Thus the characteristic property of space, that of 
having three dimensions, is only a property of our 
distribution board, a property residing, so to speak, 
in the human intelligence. The destruction of some 
of these connexions, that is to say, of these associa- 


tions of ideas, would be sufficient to give us a dif- 
ferent distribution board, and that might be enough 
to endow space with a fourth dimension. 

Some people will be astonished at such a result. 
The exterior world, they think, must surely count 
for something. If the number of dimensions comes 
from the way in which we are made, there might 
be thinking beings living in our world, but made 
differently from us, who would think that space has 
more or less than three dimensions. Has not M. 
de Cyon said that Japanese mice, having only two 
pairs of semicircular canals, think that space has 
two dimensions ? Then will not this thinking being, 
if he is capable of constructing a physical system, 
make a system of two or four dimensions, which 
yet, in a sense, will be the same as ours, since it will 
be the description of the same world in another 
language ? 

It quite seems, indeed, that it would be possible to 
translate our physics into the language of geometry 
of four dimensions. Attempting such a translation 
would be giving oneself a great deal of trouble for 
little profit, and I will content myself with men- 
tioning Hertz's mechanics, in which something of 
the kind may be seen. Yet it seems that the 
translation would always be less simple than the 
text, and that it would never lose the appearance of 
a translation, for the language of three dimensions 
seems the best suited to the description of our 
world, even though that description may be made, 
in case of necessity, in another idiom. 

Besides, it is not by chance that our distribution 
board has been formed. There is a connexion 
(1.777) 8 


between the alarm A I and the parry Bl, that is, a 
property residing in our intelh'gence. But why is 
there this connexion? It is because the parry Bi 
enables us effectively to defend ourselves against the 
danger Ai, and that is a fact exterior to us, a 
property of the exterior world. Our distribution 
board, then, is only the translation of an assemblage 
of exterior facts ; if it has three dimensions, it is 
because it has adapted itself to. a world having 
certain properties, and the most important of these 
properties is that there exist natural solids which 
are clearly displaced in accordance with the laws 
we call laws of motion of unvarying solids. If, then, 
the language of three dimensions is that which 
enables us most easily to describe our world, we 
must not be surprised. This language is founded 
on our distribution board, and it is in order to 
enable us to live in this world that this board has 
been established. 

I have said that we could conceive of thinking 
beings, living in our world, whose distribution board 
would have four dimensions, who would, consequently, 
think in hyperspace. It is not certain, however, that 
such beings, admitting that they were born, would 
be able to live and defend themselves against the 
thousand dangers by which they would be assailed. 


A few remarks in conclusion. There is a striking 
contrast between the roughness of this primitive 
geometry which is reduced to what I call a distribu- 
tion board, and the infinite precision of the geometry 
of geometricians. And yet the latter is the child ot 


the former, but not of it alone ; it required to be 
fertilized by the faculty we have of constructing 
mathematical concepts, such, for instance, as that of 
the group. It was necessary to find among these 
pure concepts the one that was best adapted to 
this rough space, whose genesis I have tried to 
explain in the preceding pages, the space which is 
common to us and the higher animals. 

The evidence of certain geometrical postulates is 
only, as I have said, our unwillingness to give up 
very old habits. But these postulates are infinitely 
precise, while the habits have about them some- 
thing essentially fluid. As soon as we wish to think, 
we are bound to have infinitely precise postulates, 
since this is the only means of avoiding contradic- 
tion. But among all the possible systems of postu- 
lates, there are some that we shall be unwilling to 
choose, because they do not accord sufficiently with 
our habits. However fluid and elastic these may be, 
they have a limit of elasticity. 

It will be seen that though geometry is not an 
experimental science, it is a science born in con- 
nexion with experience ; that we have created the 
space it studies, but adapting it to the world in 
which we live. We have chosen the most con- 
venient space, but experience guided our choice. 
As the choice was unconscious, it appears to be 
imposed upon us. Some say that it is imposed by 
experience, and others that we are born with our 
space ready-made. After the preceding considera- 
tions, it will be seen what proportion of truth and 
of error there is in these two opinions. 

In this progressive education which has resulted 


in the construction of space, it is very difficult to 
determine what is the share of the individual and 
what of the race. To what extent could one of us, 
transported from his birth into an entirely different 
world, where, for instance, there existed bodies dis- 
placed in accordance with the laws of motion of 
non-Euclidian solids — to what extent, I say, would 
he be able to give up the ancestral space in order 
to build up an entirely new space ? 

The share of the race seems to preponderate largely, 
and yet if it is to it that we owe the rough space, 
the fluid space of which I spoke just now, the space 
of the higher animals, is it not to the unconscious 
experience of the individual that we owe the in- 
finitely precise space of the geometrician? This is 
a question that is not easy of solution. I would 
mention, however, a fact which shows that the space 
bequeathed to us by our ancestors still preserves a 
certain plasticity. Certain hunters learn to shoot 
fish under the water, although the image of these 
fish is raised by refraction ; and, moreover, they do 
it instinctively. Accordingly they have learnt to 
modify their ancient instinct of direction, or, if you 
will, to substitute for the association Ai, Bi, another 
association Ai, B2, because experience has shown 
them that the former does not succeed. 



I. I have to speak here of general definitions in 
mathematics. At least that is what the title of the 
chapter says, but it will be impossible for me to 
confine myself to the subject as strictly as the rule 
of unity of action demands. I shall not be able to 
treat it without speaking to some extent of other 
allied questions, and I must ask your kind forgiveness 
if I am thus obliged from time to time to walk among 
the flower-beds to right or left. 

What is a good definition ? For the philosopher 
or the scientist, it is a definition which applies to 
all the objects to be defined, and applies only to 
them ; it is that which satisfies the rules of logic. 
But in education it is not that ; it is one that can be 
understood by the pupils. 

How is it that there are so many minds that are 
incapable of understanding mathematics? Is there 
not something paradoxical in this ? Here is a 
science which appeals only to the fundamental 
principles of logic, to the principle of contradiction, 
for instance, to what forms, so to speak, the skeleton 
of our understanding, to what we could not be de- 
prived of without ceasing to think, and yet there are 


people who find it obscure, and actually they are the 
majority. That they should be incapable of discovery 
we can understand, but that they should fail to under- 
stand the demonstrations expounded to them, that 
they should remain blind when they are shown a 
light that seems to us to shine with a pure brilliance, 
it is this that is altogether miraculous. 

And yet one need have no great experience of 
examinations to know that these blind people are 
by no means exceptional beings. We have here a 
problem that is not easy of solution, but yet must 
engage the attention of all who wish to devote them- 
selves to education. 

What is understanding ? Has the word the same 
meaning for everybody? Does understanding the 
demonstration of a theorem consist in examining each 
of the syllogisms of which it is composed in succession, 
and being convinced that it is correct and conforms 
to the rules of the game? In the same way, does 
understanding a definition consist simply in recog- 
nizing that the meaning of all the terms employed 
is already known, and being convinced that it in- 
volves no contradiction ? 

Yes, for some it is ; when they have arrived at the 
conviction, they will say, I understand. But not 
for the majority. Almost all are more exacting ; 
they want to know not only whether all the syllo- 
gisms of a demonstration are correct, but why they 
are linked together in one order rather than in 
another. As long as they appear to them engendered 
by caprice, and not by an intelligence constantly 
conscious of the end to be attained, they do not think 
they have understood. 


No doubt they are not themselves fully aware of 
what they require and could not formulate their 
desire, but if they do not obtain satisfaction, they 
feel vaguely that something is wanting. Then what 
happens? At first they still perceive the evidences 
that are placed before their eyes, but, as they are 
connected by too attenuated a thread with those that 
precede and those that follow, they pass without 
leaving a trace in their brains, and are immediately 
forgotten ; illuminated for a moment, they relapse 
at once into an eternal night. As they advance 
further, they will no longer see even this ephemeral 
light, because the theorems depend one upon another, 
and those they require have been forgotten. Thus 
it is that they become incapable of understanding 

It is not always the fault of their instructor. Often 
their intellect, which requires to perceive the connect- 
ing thread, is too sluggish to seek it and find it. But 
in order to come to their assistance, we must first of 
all thoroughly understand what it is that stops them. 

Others will always ask themselves what use it is. 
They will not have understood, unless they find 
around them, in practice or in nature, the object of 
such and such a mathematical notion. Under each 
word they wish to put a sensible image ; the definition 
must call up this image, and at each stage of the 
demonstration they must see it being transformed 
and evolved. On this condition only will they under- 
stand and retain what they have understood. These 
often deceive themselves : they do not listen to the 
reasoning, they look at the figures ; they imagine that 
they have understood when they have only seen. 


2. What different tendencies we have here ! Are 
we to oppose them, or are we to make use of them ? 
And if we wish to oppose them, which are we to 
favour? Are we to show those who content them- 
selves with the pure logic that they have only seen 
one side of the matter, or must we tell those who are 
not so easily satisfied that what they demand is not 
necessary ? 

In other words, should we constrain young people 
to change the nature of their minds ? Such an 
attempt would be useless ; we do not possess the 
philosopher's stone that would enable us to transmute 
the metals entrusted to us one into the other. All 
that we can do is to work them, accommodating our- 
selves to their properties. 

Many children are incapable of becoming mathe- 
maticians who must none the less be taught 
mathematics ; and mathematicians themselves are 
not all cast in the same mould. We have only to 
read their works to distinguish among them two kinds 
of minds — logicians like Weierstrass, for instance, and 
intuitionists like Riemann. There is the same 
difference among our students. Some prefer to treat 
their problems " by analysis," as they say, others " by 

It is quite useless to seek to change anything in 
this, and besides, it would not be desirable. It is 
well that there should be logicians and that there 
should be intuitionists. Who would venture to 
say whether he would prefer that Weierstrass had 
never written or that there had never been a Rie- 
mann ? And so we must resign ourselves to the 
diversity of minds, or rather we must be glad of it. 


3. Since the word understand has several meanings, 
the definitions that will be best understood by some 
are not those that will be best suited to others. We 
have those who seek to create an image, and those 
who restrict themselves to combining empty forms, 
perfectly intelligible, but purely intelligible, and de- 
prived by abstraction of all matter. 

I do not know whether it is necessary to quote 
any examples, but I will quote some nevertheless, 
and, first, the definition of fractions will furnish us with 
an extreme example. In the primary schools, when 
they want to define a fraction, they cut up an apple or 
a pie. Of course this is done only in imagination and 
not in reality, for I do not suppose the budget of primary 
education would allow such an extravagance. In the 
higher normal school, on the contrary, or in the 
universities, they say : a fraction is the combination 
of two whole numbers separated by a horizontal line. 
By conventions they define the operations that these 
symbols can undergo ; they demonstrate that the rules 
of these operations are the same as in the calculation 
of whole numbers ; and, lastly, they establish that 
multiplication of the fraction by the denominator, 
in accordance with these rules, gives the numerator. 
This is very well, because it is addressed to young 
people long since familiarized with the notion of 
fractions by dint of cutting up apples and other 
objects, so that their mind, refined by a considerable 
mathematical education, has, little by little, come to 
desire a purely logical definition. But what would 
be the consternation of the beginner to whom we 
attempted to offer it ? 

Such, also, are the definitions to be found in a 


book that has been justly admired and has received 
several awards of merit — Hilbert's " Grundlagen der 
Geometric." Let us see how he begins. " Imagine 
three systems of THINGS, which we will call points, 
straight lines, and planes." What these " things " are 
we do not know, and we do not need to know — it 
would even be unfortunate that we should seek to 
know ; all that we have the right to know about them 
is that we should learn their axioms, this one, for 
instance : " Two different points always determine 
a straight line," which is followed by this comment- 
ary : " Instead of determine we may say that the 
straight line passes through these two points, or that 
it joins these two points, or that the two points are 
situated on the straight line." Thus " being situated 
on a straight line" is simply defined as synonymous 
with " determining a straight line." Here is a book 
of which I think very highly, but which I should not 
recommend to a schoolboy. For the matter of that 
I might do it without fear ; he would not carry his 
reading very lar. 

I have taken extreme examples, and no instructor 
would dream of going so far. But, even though he 
comes nowhere near such models, is he not still 
exposed to the same danger? 

We are in a class of the fourth grade. The teacher 
is dictating : " A circle is the position of the points 
in a plane which are the same distance from an in- 
terior point called the centre." The good pupil writes 
this phrase in his copy-book and the bad pupil draws 
faces, but neither of them understands. Then the 
teacher takes the chalk and draws a circle on the 
board. "Ah," think the pupils, "why didn't he say 


at once, a circle is a round, and we should have 
understood." No doubt it is the teacher who is 
right. The pupils' definition would have been of no 
value, because it could not have been used for any 
demonstration, and chiefly because it could not have 
given them the salutary habit of analyzing their con- 
ceptions. But they should be made to see that they 
do not understand what they think they understand, 
and brought to realize the roughness of their primitive 
concept, and to be anxious themselves that it should 
be purified and refined. 

4. I shall return to these examples ; I only wished 
to show the two opposite conceptions. There is a 
violent contrast between them, and this contrast is 
explained by the history of the science. If we read 
a book written fifty years ago, the greater part of the 
arguments appear to us devoid of exactness. 

At that period they assumed that a continuous func- 
tion cannot change its sign without passing through 
zero, but to-day we prove it. They assumed that the 
ordinary rules of calculus are applicable to incommen- 
surable numbers ; to-day we prove it. They assumed 
many other things that were sometimes untrue. 

They trusted to intuition, but intuition cannot give 
us exactness, nor even certainty, and this has been 
recognized more and more. It teaches us, for instance, 
that every curve has a tangent — that is to say, that 
every continuous function has a derivative — and that 
is untrue. As certainty was required, it has been 
necessary to give less and less place to intuition. 

How has this necessary evolution come about ? It 
was not long before it was recognized that exactness 


cannot be established in the arguments unless it is 
first introduced into the definitions. 

For a long time the objects that occupied the atten- 
tion of mathematicians were badly defined. They 
thought they knew them because they represented 
them by their senses or their imagination, but they 
had only a rough image, and not a precise idea such 
as reasoning can take hold of. 

It is to this that the logicians have had to apply their 
efforts, and similarly for incommensurable numbers. 

The vague idea of continuity which we owe to 
intuition has resolved itself into a complicated system 
of inequalities bearing on whole numbers. Thus it 
is that all those difficulties which terrified our ances- 
tors when they reflected upon the foundations of the 
infinitesimal calculus have finally vanished. 

In analysis to-day there is no longer anything but 
whole numbers, or finite or infinite systems of whole 
numbers, bound together by a network of equalities 
and inequalities. Mathematics, as it has been said, 
has been arithmetized. 

5. But we must not imagine that the science of 
mathematics has attained to absolute exactness with- 
out making any sacrifice. What it has gained in 
exactness it has lost in objectivity. It is by with- 
drawing from reality that it has acquired this perfect 
purity. We can now move freely over its whole 
domain, which formerly bristled with obstacles. But 
these obstacles have not disappeared ; they have only 
been removed to the frontier, and will have to be 
conquered again if we wish to cross the frontier and 
penetrate into the realms of practice. 


We used to possess a vague notion, formed of in- 
congruous elements, some a priori and others derived 
from more or less digested experiences, and we im- 
agined we knew its principal properties by intuition. 
To-day we reject the empirical element and preserve 
only the a priori ones. One of the properties 
serves as definition, and all the others are de- 
duced from it by exact reasoning. This is very well, 
but it still remains to prove that this property, which 
has become a definition, belongs to the real objects 
taught us by experience, from which we had drawn 
our vague intuitive notion. In order to prove it we 
shall certainly have to appeal to experience or make 
an effort of intuition ; and if we cannot prove it, our 
theorems will be perfectly exact but perfectly useless. 

Logic sometimes breeds monsters. For half a 
century there has been springing up a host of weird 
functions, which seem to strive to have as little resem- 
blance as possible to honest functions that are of some 
use. No more continuity, or else continuity but no 
derivatives, etc. More than this, from the point of 
view of logic, it is these strange functions that are 
the most general ; those that are met without being 
looked for no longer appear as more than a particular 
case, and they have only quite a little corner left them. 

Formerly, when a new function was invented, it 
was in view of some practical end. To-day they are 
invented on purpose to show our ancestors' reasonings 
at fault, and we shall never get anything more than 
that out of them. 

If logic were the teacher's only guide, he would 
have to begin with the most general, that is to say, 
with the most weird, functions. He would have to 


set the beginner to wrestle with this collection of 
monstrosities. If you don't do so, the logicians might 
say, you will only reach exactness by stages. 

6. Possibly this may be true, but we cannot take 
such poor account of reality, and I do not mean 
merely the reality of the sensible world, which has 
its value nevertheless, since it is for battling with 
it that nine-tenths of our pupils are asking for arms. 
There is a more subtle reality which constitutes the 
life of mathematical entities, and is something more 
than logic. 

Our body is composed of cells, and the cells of 
atoms, but are these cells and atoms the whole reality 
of the human body? Is not the manner in which 
these cells are adjusted, from which results the unity 
of the individual, also a reality, and of much greater 
interest ? 

Would a naturalist imagine that he had an adequate 
knowledge of the elephant if he had never studied the 
animal except through a microscope? 

It is the same in mathematics. When the logician 
has resolved each demonstration into a host of ele- 
mentary operations, all of them correct, he will not yet 
be in possession of the whole reality ; that indefinable 
something that constitutes the unity of the demonstra- 
tion will still escape him completely. 

What good is it to admire the mason's work in the 
edifices erected by great architects, if we cannot under- 
stand the general plan of the master ? Now pure logic 
cannot give us this view of the whole ; it is to intuition 
we must look for it. 

Take, for instance, the idea of the continuous func- 


tion. To begin with, it is only a perceptible image, 
a line drawn with chalk on a blackboard. Little by 
little it is purified ; it is used for constructing a com- 
plicated system of inequalities which reproduces all 
the lines of the original image ; when the work is 
quite finished, the centering is removed, as it is after 
the construction of an arch ; this crude representation 
is henceforth a useless support, and disappears, and 
there remains only the edifice itself, irreproachable in 
the eyes of the logician. And yet, if the instructor 
did not recall the original image, if he did not replace 
the centering for a moment, how would the pupil 
guess by what caprice all these inequalities had been 
scaffolded in this way one upon another? The defini- 
tion would be logically correct, but it would not show 
him the true reality. 

7. And so we are obliged to make a step back- 
wards. No doubt it is hard for a master to teach 
what does not satisfy him entirely, but the satisfaction 
of the master is not the sole object of education. We 
have first to concern ourselves with the pupil's state 
of mind, and what we want it to become. 

Zoologists declare that the embryonic development 
of an animal repeats in a very short period of time 
the whole history of its ancestors of the geological 
ages. It seems to be the same with the development 
of minds. The educator must make the child pass 
through all that his fathers have passed through, more 
rapidly, but without missing a stage. On this account, 
the history of any science must be our first guide. 

Our fathers imagined they knew what a fraction 
was, or continuity, or the area of a curved surface ; it 


is we who have reahzed that they did not. In the 
same way our pupils imagine that they know it when 
they begin to study mathematics seriously. If, with- 
out any other preparation, I come and say to them : 
" No, you do not know it ; you do not understand 
what you imagine you understand ; I must demon- 
strate to you what appears to you evident ; " and if, 
in the demonstration, I rely on premises that seem 
to them less evident than the conclusion, what will 
the wretched pupils think ? They will think that the 
science of mathematics is nothing but an arbitrary 
aggregation of useless subtleties ; or they will lose 
their taste for it ; or else they will look upon it as 
an amusing game, and arrive at a state of mind 
analogous to that of the Greek sophists. 

Later on, on the contrary, when the pupil's mind 
has been familiarized with mathematical reasoning 
and ripened by this long intimacy, doubts will spring 
up of their own accord, and then your demonstration 
will be welcome. It will arouse new doubts, and 
questions will present themselves successively to the 
child, as they presented themselves successively to 
our fathers, until they reach a point when only perfect 
exactness will satisfy them. It is not enough to feel 
doubts about everything; we must know why we doubt. 

8. The principal aim of mathematical education is 
to develop certain faculties of the mind, and among 
these intuition is not the least precious. It is through 
it that the mathematical world remains in touch with 
the real world, and even if pure mathematics could 
do without it, we should still have to have recourse 
to it to fill up the gulf that separates the symbol 


from reality. The practitioner will always need it, 
and for every pure geometrician there must be a 
hundred practitioners. 

The engineer must receive a complete mathematical 
training, but of what use is it to be to him, except to 
enable him to see the different aspects of things and 
to see them quickly ? He has no time to split hairs. 
In the complex physical objects that present them- 
selves to him, he must promptly recognize the point 
where he can apply the mathematical instruments we 
have put in his hands. How could he do this if we left 
between the former and the latter that deep gulf dug 
by the logicians ? 

9. Beside the future engineers are other less numerous 
pupils, destined in their turn to become teachers, and 
so they must go to the very root of the matter ; a 
profound and exact knowledge of first principles is 
above all indispensable for them. But that is no 
reason for not cultivating their intuition, for they 
would form a wrong idea of the science if they never 
looked at it on more than one side, and, besides, they 
could not develop in their pupils a quality they did 
not possess themselves. 

For the pure geometrician hintself this faculty is 
necessary : it is by logic that we prove, but by intui- 
tion that we discover. To know how to criticize is 
good, but to know how to create is better. You 
know how to recognize whether a combination is 
correct, but much use this will be if you do not 
possess the art of selecting among all the possible 
combinations. Logic teaches us that on such and 
such a road we are sure of not meeting an obstacle ; 

(1,777) 9 


it does not tell us which is the road that leads to the 
desired end. For this it is necessary to see the end 
from afar, and the faculty which teaches us to see is 
intuition. Without it, the geometrician would be like 
a writer well up in grammar but destitute of ideas. 
Now how is this faculty to develop, if, as soon as it 
shows itself, it is hounded out and proscribed, if we 
learn to distrust it before we know what good can be 
got from it ? 

And here let me insert a parenthesis to insist on 
the importance of written exercises. Compositions 
in writing are perhaps not given sufficient prominence 
in certain examinations. In the 'kcole Poly technique, for 
instance, I am told that insistence on such compositions 
would close the door to very good pupils who know 
their subject and understand it very well, and yet are 
incapable of applying it in the smallest degree. I 
said just above that the word understand has several 
meanings. Such pupils only understand in the first 
sense of the word, and we have just seen that this 
is not sufficient to make either an engineer or a 
geometrician. Well, since we have to make a choice, 
I prefer to choose those who understand thoroughly. 

lo. But is not the art of exact reasoning also a 
precious quality that the teacher of mathematics 
should cultivate above all else? I am in no danger 
of forgetting it : we must give it attention, and that 
from the beginning. I should be distressed to see 
geometry degenerate into some sort of low - grade 
tachymetrics, and I do not by any means subscribe 
to the extreme doctrines of certain German professors. 
But we have sufficient opportunity of training pupils 


in correct reasoning in those parts of mathematics in 
which the disadvantages I have mentioned do not 
occur. We have long series of theorems in which 
absolute logic has ruled from the very start and, so to 
speak, naturally, in which the first geometricians have 
given us models that we must continually imitate and 

It is in expounding the first principles that we must 
avoid too much subtlety, for there it would be too 
disheartening, and useless besides. We cannot prove 
everything, we cannot define everything, and it will 
always be necessary to draw upon intuition. What 
does it matter whether we do this a little sooner or a 
little later, and even whether we ask for a little more 
or a little less, provided that, making a correct use 
of the premises it gives us, we learn to reason 
accurately ? 

ir. Is it possible to satisfy so many opposite 
conditions? Is it possible especially when it is a 
question of giving a definition ? How are we to find 
a statement that will at the same time satisfy the 
inexorable laws of logic and our desire to understand 
the new notion's place in the general scheme of the 
science, our need of thinking in images ? More often 
than not we shall not find it, and that is why the 
statement of a definition is not enough ; it must be 
prepared and it must be justified. 

What do I mean by this ? You know that it has 
often been said that every definition implies an axiom, 
since it asserts the existence of the object defined. 
The definition, then, will not be justified, from the 
purely logical point of view, until we have proved that 


it involves no contradiction either in its terms or with 
the truths previously admitted. 

But that is not enough. A definition is stated as 
a convention, but the majority of minds will revolt 
if you try to impose it upon them as an arbitrary 
convention. They will have no rest until you have 
answered a great number of questions. 

Mathematical definitions are most frequently, as 
M. Liard has shown, actual constructions built up 
throughout of simpler notions. But why should these 
elements have been assembled in this manner, when 
a thousand other assemblages were possible ? Is it 
simply caprice? If not, why had this combination 
more right to existence than any of the others ? What 
need does it fill ? How was it foreseen that it would 
play an important part in the development of the 
science, that it would shorten our reasoning and our 
calculations? Is there any familiar object in nature 
that is, so to speak, its indistinct and rough image? 

That is not all. If you give a satisfactory answer 
to all these questions, we shall realize that the new- 
comer had the right to be baptized. But the choice of 
a name is not arbitrary either ; we must explain what 
analogies have guided us, and that if we have given 
analogous names to different things, these things at 
least differ only in matter, and have some resemblance 
in form, that their properties are analogous and, so to 
speak, parallel. 

It is on these terms that we shall satisfy all propen- 
sities. If the statement is sufficiently exact to please 
the logician, the justification will satisfy the intui- 
tionist. But we can do better still. Whenever it is 
possible, the justification will precede the statement 


and prepare it. The general statement will be led up 
to by the study of some particular examples. 

One word more. The aim of each part of the 
statement of a definition is to distinguish the object 
to be defined from a class of other neighbouring 
objects. The definition will not be understood until 
you have shown not only the object defined, but the 
neighbouring objects from which it has to be dis- 
tinguished, until you have made it possible to grasp 
the difference, and have added explicitly your reason 
for saying this or that in stating the definition. 

But it is time to leave generalities and to enquire 
how the somewhat abstract principles I have been 
expounding can be applied in arithmetic, in geometry, 
in analysis, and in mechanics. 


12. We do not have to define the whole number. 
On the other hand, operations on whole numbers are 
generally defined, and I think the pupils learn these 
definitions by heart and attach no meaning to them. 
For this there are two reasons : first, they are taught 
them too early, while their mind still feels no need of 
them ; and then these definitions are not satisfactory 
from the logical point of view. For addition, we 
cannot find a good one, simply because we must 
stop somewhere, and cannot define everything. The 
definition of addition is to say that it consists in adding. 
All that we can do is to start with a certain number 
of concrete examples and say, the operation that has 
just been performed is called addition. 

For subtraction it is another matter. It can be 
defined logically as the inverse operation of addition. 


But is that how we should begin ? Here, again, we 
should start with examples, and show by these 
examples the relation of the two operations. Thus 
the definition will be prepared and justified. 

In the same way for multiplication. We shall take 
a particular problem ; we shall show that it can be 
solved by adding several equal numbers together ; 
we shall then point out that we arrive at the result 
quicker by multiplication, the operation the pupils 
perform already by rote, and the logical definition will 
spring from this quite naturally. 

We shall define division as the inverse operation 
of multiplication ; but we shall begin with an example 
drawn from the familiar notion of sharing, and we 
shall show by this example that multiplication 
reproduces the dividend. 

There remain the operations on fractions. There is 
no difficulty except in the case of multiplication. The 
best way is first to expound the theory of proportions, 
as it is from it alone that the logical definition can 
spring. But, in order to gain acceptance for the 
definitions that are met with at the start in this theory, 
we must prepare them by numerous examples drawn 
from classical problems of the rule of three, and we 
shall be careful to introduce fractional data. We shall 
not hesitate, either, to familiarize the pupils with the 
notion of proportion by geometrical figures ; either 
appealing to their recollection if they have already 
done any geometry, or having recourse to direct 
intuition if they have not, which, moreover, will prepare 
them to do it. I would add, in conclusion, that after 
having defined the multiplication of fractions, we must 
justify this definition by demonstration that it is 


commutative, associative, and distributive, making it 
quite clear to the listeners that the verification has 
been made in order to justify the definition. 

We see what part is played in all this by geometrical 
figures, and this part is justified by the philosophy and 
the history of the science. If arithmetic had remained 
free from all intermixture with geometry, it would 
never have known anything but the whole number. 
It was in order to adapt itself to the requirements of 
geometry that it discovered something else. 


In geometry we meet at once the notion of the 
straight line. Is it possible to define the straight 
line ? The common definition, the shortest path from 
one point to another, does not satisfy me at all. I 
should start simply with the ruler, and I should first 
show the pupil how we can verify a ruler by revolving 
it. This verification is the true definition of a straight 
line, for a straight line is an axis of rotation. We 
should then show him how to verify the ruler by 
sliding it, and we should have one of the most im- 
portant properties of a straight line. As for that 
other property, that of being the shortest path from 
one point to another, it is a theorem that can be 
demonstrated apodeictically, but the demonstration is 
too advanced to find a place in secondary education. 
It will be better to show that a ruler previously veri- 
fied can be applied to a taut thread. We must not 
hesitate, in the presence of difficulties of this kind, 
to multiply the axioms, justifying them by rough 

Some axioms we must admit ; and if we admit a 


few more than is strictly necessary, the harm is not 
great. The essential thing is to learn to reason 
exactly with the axioms once admitted. Uncle 
Sarcey, who loved to repeat himself, often said that 
the audience at a theatre willingly accepts all the 
postulates imposed at the start, but that once the 
curtain has gone up it becomes inexorable on the 
score of logic. Well, it is just the same in mathe- 

For the circle we can start with the compass. The 
pupils will readily recognize the curve drawn. We 
shall then point out to them that the distance of the 
two points of the instrument remains constant, that 
one of these points is fixed and the other movable, 
and we shall thus be led naturally to the logical 

The definition of a plane implies an axiom, and 
we must not attempt to conceal the fact. Take a 
drawing-board and point out how a movable ruler 
can be applied constantly to the board, and that 
while still retaining three degrees of freedom. We 
should compare this with the cylinder and the cone, 
surfaces to which a straight line cannot be applied 
unless we allow it only two degrees of freedom. 
Then we should take three drawing-boards, and we 
should show first that they can slide while still re- 
maining in contact with one another, and that with 
three degrees of freedom. And lastly, in order to 
distinguish the plane from the sphere, that two of 
these boards that can be applied to a third can also 
be applied to one another. 

Perhaps you will be surprised at this constant use 
of movable instruments. It is not a rough artifice. 


and it is much more philosophical than it would 
appear at first sight. What is geometry for the 
philosopher ? It is the study of a group. And what 
group ? That of the movements of solid bodies. How 
are we to define this group, then, without making some 
solid bodies move? 

Are we to preserve the classical definition of par- 
allels, and say that we give this name to two straight 
lines, situated in the same plane, which, being pro- 
duced ever so far, never meet? No, because this 
definition is negative, because it cannot be verified 
by experience, and cannot consequently be regarded 
as an immediate datum of intuition, but chiefly because 
it is totally foreign to the notion of group and to the 
consideration of the motion of solid bodies, which is, 
as I have said, the true source of geometry. Would 
it not be better to define first the rectilineal trans- 
position of an invariable figure as a motion in which 
all the points of this figure have rectilineal trajectories, 
and to show that such a transposition is possible, 
making a square slide on a ruler ? From this experi- 
mental verification, raised to the form of an axiom, 
it would be easy to educe the notion of parallel and 
Euclid's postulate itself. 


I need not go back to the definition of velocity or 
of acceleration or of the other kinematic notions : 
they will be more properly connected with ideas of 
space and time, which alone they involve. 

On the contrary, I will dwell on the dynamic 
notions of force and mass. 

There is one thing that strikes me, and that is, how 


far young people who have received a secondary 
education are from applying the mechanical laws 
they have been taught to the real world. It is not 
only that they are incapable of doing so, but they 
do not even think of it. For them the world of 
science and that of reality are shut off in water-tight 
compartments. It is not uncommon to see a well- 
dressed man, probably a university man, sitting in 
a carriage and imagining that he is helping it on by 
pushing on the dash-board, and that in disregard of 
the principle of action and reaction. 

If we try to analyze the state of mind of our pupils, 
this will surprise us less. What is for them the true 
definition of force ? Not the one they repeat, but the 
one that is hidden away in a corner of their intellect, 
and from thence directs it all. This is their definition: 
Forces are arrows that parallelograms are made of; 
these arrows are imaginary things that have nothing 
to do with anything that exists in nature. This would 
not happen if they were shown forces in reality before 
having them represented by arrows. 

How are we to define force? If we want a logical 
definition, there is no good one, as I think I have 
shown satisfactorily elsewhere. There is the anthro- 
pomorphic definition, the sensation of muscular effort ; 
but this is really too crude, and we cannot extract 
anything useful from it. 

This is the course we ought to pursue. First, in 
order to impart a knowledge of the genus force, we 
must show, one after the other, all the species of this 
genus. They are very numerous and of great variety. 
There is the pressure of liquids on the sides of the 
vessels in which they are contained, the tension of 


cords, the elasticity of a spring, gravity that acts on 
all the molecules of a body, friction, the normal 
mutual action and reaction of two solids in contact. 

This is only a qualitative definition ; we have to 
learn to measure a force. For this purpose we shall 
show first that we can replace one force by another 
without disturbing the equilibrium, and we shall find 
the first example of this substitution in the balance 
and Borda's double scales. Then we shall show that 
we can replace a weight not only by another weight, 
but by forces of different nature ; for example, Prony's 
dynamometer break enables us to replace a weight 
by friction. 

From all this arises the notion of the equivalence 
of two forces. 

We must also define the direction of a force. If 
a force F is equivalent to another force F^ that is 
applied to the body we are dealing with through the 
medium of a taut cord, in such a way that F can be 
replaced by F-^ without disturbing the equilibrium, 
then the point of attachment of the cord will be, by 
definition, the point of application of the force F^ and 
that of the equivalent force F, and the direction of the 
cord will be the direction of the force F-^ and also that 
of the equivalent force F. 

From this we shall pass to the comparison of the 
magnitude of forces. If one force can replace two 
others of the same direction, it must be equal to their 
sum, and we shall show, for instance, that a weight of 
20 ounces can replace two weights of 10 ounces. 

But this is not all. We know now how to compare 
the intensity of two forces which have the same direc- 
tion and the same point of application, but we have 


to learn to do this when the directions are different. 
For this purpose we imagine a cord stretched by a 
weight and passing over a pulley ; we say that the 
tension of the two portions of the cord is the same, 
and equal to the weight 

Here is our definition. It enables us to compare 
the tensions of our two portions, and, by using the 
preceding definitions, to compare two forces of any 
kind having the same direction as these two portions. 
We have to justify it by showing that the tension of 
the last portion remains the same for the same weight, 
whatever be the number and the disposition of the 
pulleys. We must then complete it by showing that 
this is not true unless the pulleys are without friction. 

Once we have mastered these definitions we must 
show that the point of application, the direction, and 
the intensity are sufficient to determine a force ; that 
two forces for which these three elements are the same 
are always equivalent, and can always be replaced one 
by the other, either in equilibrium or in motion, and 
that whatever be the other forces coming into play. 

We must show that two concurrent forces can 
always be replaced by a single resultant force, and 
that this resultant remains the same whether the body 
is in repose or in motion, and whatever be the other 
forces applied to it. 

Lastly, we must show that forces defined as we have 
defined them satisfy the principle of the equality of 
action and reaction. 

All this we learn by experiment, and by experiment 

It will be sufficient to quote some common experi- 
ments that the pupils make every day without being 


aware of it, and to perform before them a small 
number of simple and well-selected experiments. 

It is not until we have passed through all these 
roundabout ways that we can represent forces by 
arrows, and even then I think it would be well, from 
time to time, as the argument develops, to come back 
from the symbol to the reality. It would not be 
difficult, for instance, to illustrate the parallelogram 
of forces with the help of an apparatus composed of 
three cords passing over pulleys, stretched by weights, 
and producing equilibrium by pulling on the same 

Once we know force, it is easy to define mass. 
This time the definition must be borrowed from 
dynamics. We cannot do otherwise, since the end 
in view is to make clear the distinction between mass 
and weight. Here, again, the definition must be pre- 
pared by experiments. There is, indeed, a machine 
that seems to be made on purpose to show what 
mass is, and that is Atwood's machine. Besides this 
we shall recall the laws of falling bodies, and how 
acceleration of gravity is the same for heavy as for 
light bodies, and varies according to latitude, etc. 

Now if you tell me that all the methods I advocate 
have long since been applied in schools, I shall be 
more pleased than surprised to hear it. I know that 
on the whole our mathematical education is good ; I 
do not wish to upset it, and should even be distressed 
at this result ; I only desire gradual, progressive im- 
provements. This education must not undergo sudden 
variations at the capricious breath of ephemeral fashions. 
In such storms its high educative value would soon 
founder. A good and sound logic must continue to 


form its toundation. Definition by example is always 
necessary, but it must prepare the logical definition 
and not take its place ; it must at least make its want 
felt in cases where the true logical definition cannot be 
given to any purpose except in higher education. 

You will understand that what I have said here in 
no sense implies the abandonment of what I have 
written elsewhere. I have often had occasion to 
criticize certain definitions which I advocate to-day. 
These criticisms hold good in their entirety ; the 
definitions can only be provisional, but it is through 
them that we must advance. 




Can mathematics be reduced to logic without having 
to appeal to principles peculiar to itself? There is a 
whole school full of ardour and faith who make it 
their business to establish the possibility. They have 
their own special language, in which words are used 
no longer, but only signs. This language can be 
understood only by the few initiated, so that the 
vulgar are inclined to bow before the decisive affirma- 
tions of the adepts. It will, perhaps, be useful to 
examine these affirmations somewhat more closely, in 
order to see whether they justify the peremptory tone 
in which they are made. 

But in order that the nature of the question should 
be properly understood, it is necessary to enter into 
some historical details, and more particularly to review 
the character of Cantor's work. 

The notion of infinity had long since been introduced 
into mathematics, but this infinity was what philoso- 
phers call a becoming. Mathematical infinity was only 
a quantity susceptible of growing beyond all limit ; it 
was a variable quantity of which it could not be said 
that it had passed, but only that it would pass, all limits. 


Cantor undertook to introduce into mathematics an 
actual infinity — that is to say, a quantity which is not 
only susceptible of passing all limits, but which is 
regarded as having already done so. He set himself 
such questions as these : Are there more points in 
space than there are whole numbers ? Are there more 
points in space than there are points in a plane ? etc. 

Then the number of whole numbers, that of points 
in space, etc., constitutes what he terms a transfinite 
cardinal number — that is to say, a cardinal number 
greater than all the ordinary cardinal numbers. And 
he amused himself by comparing these transfinite car- 
dinal numbers, by arranging in suitable order the 
elements of a whole which contains an infinite number 
of elements ; and he also imagined what he terms 
transfinite ordinal numbers, on which I will not dwell 

Many mathematicians have followed in his tracks, 
and have set themselves a series of questions of the 
same kind. They have become so familiar with trans- 
finite numbers that they have reached the point of 
making the theory of finite numbers depend on that 
of Cantor's cardinal numbers. In their opinion, if we 
wish to teach arithmetic in a truly logical way, we 
ought to begin by establishing the general properties 
of the transfinite cardinal numbers, and then distin- 
guish from among them quite a small class, that of the 
ordinary whole numbers. Thanks to this roundabout 
proceeding, we might succeed in proving all the propo- 
sitions relating to this small class (that is to say, our 
whole arithmetic and algebra) without making use of 
a single principle foreign to logic. 

This method is evidently contrary to all healthy 


psychology. It is certainly not in this manner that 
the human mind proceeded to construct mathematics, 
and I imagine, too, its authors do not dream of intro- 
ducing it into secondary education. But is it at least 
logical, or, more properly speaking, is it accurate? 
We may well doubt it. 

Nevertheless, the geometricians who have employed 
it are very numerous. They have accumulated formulas 
and imagined that they rid themselves of all that is not 
pure logic by writing treatises in which the formulas 
are no longer interspersed with explanatory text, as in 
the ordinary works on mathematics, but in which the 
text has disappeared entirely. 

Unfortunately, they have arrived at contradictory 
results, at what are called the Cantorian antinomies, 
to which we shall have occasion to return. These 
contradictions have not discouraged them, and they 
have attempted to modify their rules, in order to 
dispose of those that had already appeared, but with- 
out gaining any assurance by so doing that no new 
ones would appear. 

It is time that these exaggerations were treated as 
they deserve. I have no hope of convincing these 
logicians, for they have lived too long in this atmo- 
sphere. Besides, when we have refuted one of their 
demonstrations, we are quite sure to find it cropping 
up again with insignificant changes, and some of them 
have already risen several times from their ashes. 
Such in old times was the Lernaan hydra, with its 
famous heads that always grew again. Hercules was 
successful because his hydra had only nine heads 
(unless, indeed, it was eleven), but in this case there are 
too many, they are in England, in Germany, in Italy, 

(1.777) 10 


and in France, and he would be forced to abandon the 
task. And so I appeal only to unprejudiced people of 
common sense. 


In these latter years a large number of works have 
been published on pure mathematics and the philosophy 
of mathematics, with a view to disengaging and isolat- 
ing the logical elements of mathematical reasoning. 
These works have been analyzed and expounded 
very lucidly by M. Couturat in a work entitled 
" Les Principes des Math^matiques." 

In M. Couturat's opinion the new works, and more 
particularly those of Mr. Russell and Signor Peano, 
have definitely settled the controversy so long in 
dispute between Leibnitz and Kant. They have 
shown that there is no such thing as an a priori 
synthetic judgment (the term employed by Kant to 
designate the judgments that can neither be demon- 
strated analytically, nor reduced to identity, nor 
established experimentally); they have shown that 
mathematics is entirely reducible to logic, and that 
intuition plays no part in it whatever. 

This is what M. Couturat sets forth in the work I 
have just quoted. He also stated the same opinions 
even more explicitly in his speech at Kant's jubilee; 
so much so that I overheard my neighbour whisper : 
" It's quite evident that this is the centenary of Kant's 

Can we subscribe to this decisive condemnation ? 
I do not think so, and I will try to show why. 



What strikes us first of all in the new mathennatics 
is its purely formal character. " Imagine," says Hilbert, 
"three kinds of things, which we will call points, 
straight lines, and planes ; let us agree that a straight 
line shall be determined by two points, and that, in- 
stead of saying that this straight line is determined by 
these two points, we may say that it passes through 
these two points, or that these two points are situated 
on the straight line." What these things are, not only 
do we not know, but we must not seek to know. It is 
unnecessary, and any one who had never seen either a 
point or a straight line or a plane could do geometry 
just as well as we can. In order that the words /a.yj 
through or the words be situated on should not call up 
any image in our minds, the former is merely regarded 
as the synonym of be determined^ and the latter of 

Thus it will be readily understood that, in order to 
demonstrate a theorem, it is not necessary or even 
useful to know what it means. We might replace 
geometry by the reasoning piano imagined by Stanley 
Jevons ; or, if we prefer, we might imagine a machine 
where we should put in axioms at one end and take 
out theorems at the other, like that legendary machine 
in Chicago where pigs go in alive and come out trans- 
formed into hams and sausages. It is no more neces- 
sary for the mathematician than it is for these machines 
to know what he is doing. 

I do not blame Hilbert for this formal character of 
his geometry. He was bound to tend in this direction, 
given the problem he set himself. He wished to reduce 


to a minimum the number of the fundamental axioms 
of geometry, and to make a complete enumeration of 
them. Now, in the arguments in which our mind 
remains active, in those in which intuition still plays 
a part, in the living arguments, so to speak, it is 
difficult not to introduce an axiom or a postulate that 
passes unnoticed. Accordingly, it was not till he had 
reduced all geometrical arguments to a purely me- 
chanical form that he could be certain of having 
succeeded in his design and accomplished his work. 

What Hilbert had done for geometry, others have 
tried to do for arithmetic and analysis. Even if they 
had been entirely successful, would the Kantians be 
finally condemned to silence ? Perhaps not, for it is 
certain that we cannot reduce mathematical thought 
to an empty form without mutilating it. Even admit- 
ting that it has been established that all theorems can 
be deduced by purely analytical processes, by simple 
logical combinations of a finite number of axioms, and 
that these axioms are nothing but conventions, the 
philosopher would still retain the right to seek the 
origin of these conventions, and to ask why they were 
iudged preferable to the contrary conventions. 

And, further, the logical correctness of the argu- 
ments that lead from axioms to theorems is not the 
only thing we have to attend to. Do the rules of 
perfect logic constitute the whole of mathematics? 
As well say that the art of the chess-player reduces 
itself to the rules for the movement of the pieces. 
A selection must be made out of all the construc- 
tions that can be combined with the materials 
furnished by logic. The true geometrician makes 
this selection judiciously, because he is guided by 


a sure instinct, or by some vague consciousness of 
I know not what profounder and more hidden geom- 
etry, which alone gives a value to the constructed 

To seek the origin of this instinct, and to study 
the laws of this profound geometry which can be 
felt but not expressed, would be a noble task for 
the philosophers who will not allow that logic is 
all. But this is not the point of view I wish to 
take, and this is not the way I wish to state 
the question. This instinct I have been speaking 
of is necessary to the discoverer, but it seems at 
first as if we could do without it for the study of 
the science once created. Well, what I want to find 
out is, whether it is true that once the principles of 
logic are admitted we can, I will not say discover, 
but demonstrate all mathematical truths without 
making a fresh appeal to intuition. 


To this question I formerly gave a negative answer. 
(See " Science et Hypothese," Chapter I.) Must our 
answer be modified by recent works ? I said no, 
because " the principle of complete induction " ap- 
peared to me at once necessary to the mathematician, 
and irreducible to logic. We know the statement of 
the principle : " If a property is true of the number 
I, and if it is established that it is true of n+i pro- 
vided it is true of n, it will be true of all whole 
numbers." I recognized in this the typical mathe- 
matical argument. I did not mean to say, as has 
been supposed, that all mathematical arguments can 
be reduced to an application of this principle. 


Examining these arguments somewhat closely, we 
should discover the application of many other similar 
principles, offering the same essential characteristics. 
In this category of principles, that of complete induc- 
tion is only the simplest of all, and it is for that 
reason that I selected it as a type. 

The term principle of complete induction which 
has been adopted is not justifiable. This method 
of reasoning is none the less a true mathematical 
induction itself, which only differs from the ordinary 
induction by its certainty. 


Definitions and Axioms. 

The existence of such principles is a difficulty for 
the inexorable logicians. How do they attempt to 
escape it? The principle of complete induction, they 
say, is not an axiom properly so called, or an a 
prion synthetic judgment ; it is simply the defini- 
tion of the whole number. Accordingly it is a mere 
convention. In order to discuss this view, it will be 
necessary to make a close examination of the rela- 
tions between definitions and axioms. 

We will first refer to an article by M. Couturat 
on mathematical definitions which appeared in 
r Enseignement MatMmatique, a review published by 
Gauthier-Villars and by Georg in Geneva. We find 
a distinction between direct definition and definition 
by postulates. 

" Definition by postulates," says M. Couturat, 
" applies not to a single notion, but to a system of 
notions ; it consists in enumerating the fundamental 


relations that unite them, which make it possible to 
demonstrate all their other properties : these relations 
are postulates . . ." 

If we have previously defined all these notions 
with one exception, then this last will be by defini- 
tion the object which verifies these postulates. 

Thus certain indemonstrable axioms of mathe- 
matics would be nothing but disguised definitions. 
This point of view is often legitimate, and I have 
myself admitted it, for instance, in regard to Euclid's 

The other axioms of geometry are not sufficient to 
define distance completely. Distance, then, will be 
by definition, the one among all the magnitudes 
which satisfy the other axioms, that is of such a 
nature as to make Euclid's postulate true. 

Well, the logicians admit for the principle of com- 
plete induction what I admit for Euclid's postulate, 
and they see nothing in it but a disguised definition. 

But to give us this right, there are two conditions 
that must be fulfilled. John Stuart Mill used to say 
that every definition implies an axiom, that in which 
we affirm the existence of the object defined. On 
this score, it would no longer be the axiom that 
might be a disguised definition, but, on the contrary, 
the definition that would be a disguised axiom. 
Mill understood the word existence in a material 
and empirical sense ; he meant that in defining a 
circle we assert that there are round things in 

In this form his opinion is inadmissible. Mathe- 
matics is independent of the existence of material 
objects. In mathematics the word exist can only 


have one meaning ; it signifies exemption from 
contradiction. Thus rectified, Mill's thought becomes 
accurate. In defining an object, we assert that the 
definition involves no contradiction. 

If, then, we have a system of postulates, and if we 
can demonstrate that these postulates involve no 
contradiction, we shall have the right to consider 
them as representing the definition of one of the 
notions found among them. If we cannot demon- 
strate this, we must admit it without demonstration, 
and then it will be an axiom. So that if we wished 
to find the definition behind the postulate, we should 
discover the axiom behind the definition. 

Generally, for the purpose of showing that a 
definition does not involve any contradiction, we 
proceed by example, and try to form an example of 
an object satisfying the definition. Take the case 
of a definition by postulates. We wish to define a 
notion A, and we say that, by definition, an A is 
any object for which certain postulates are true. If 
we can demonstrate directly that all these postulates 
are true of a certain object B, the definition will be 
justified, and the object B will be an example of A. 
We shall be certain thc^t the postulates are not 
contradictory, since there are cases in which they 
are all true at once. 

But such a direct demonstration by example is 
not always possible. Then, in order to establish 
that the postulates do not involve contradiction, we 
must picture all the propositions that can be de- 
duced from these postulates considered as premises, 
and show that among these propositions there are 
no two of which one is the contradiction of the 


other. If the number of these propositions is finite, 
a direct verification is possible ; but this is a case 
that is not frequent, and, moreover, of little interest. 

If the number of the propositions is infinite, we 
can no longer make this direct verification. We 
must then have recourse to processes of demonstra- 
tion, in which we shall generally be forced to invoke 
that very principle of complete induction that we are 
attempting to verify. 

I have just explained one of the conditions which 
the logicians were bound to satisfy, and we shall see 
further on that tliey have not done so. 


There is a second condition. When we give a 
definition, it is for the purpose of using it. 

Accordingly, we shall find the word defined in the 
text that follows. Have we the right to assert, of 
the object represented by this word, the postulate 
that served as definition ? Evidently we have, if the 
word has preserved its meaning, if we have not 
assigned it a different meaning by implication. Now 
this is what sometimes happens, and it is generally 
difficult to detect it. We must see how the word 
was introduced into our text, and whether the door 
through which it came does not really imply a 
different definition from the one enunciated. 

This difficulty is encountered in all applications of 
mathematics. The mathematical notion has received 
a highly purified and exact definition, and for the 
pure mathematician all hesitation has disappeared. 
But when we come to apply it, to the physical 
sciences, for instance, we are no longer dealing with 


this pure notion, but with a concrete object which is 
often only a rough image of it. To say that this 
object satisfies the definition, even approximately, is 
to enunciate a new truth, which has no longer the 
character of a conventional postulate, and that expe- 
rience alone can establish beyond a doubt. 

But, without departing from pure mathematics, we 
still meet with the same difficulty. You give a 
subtle definition of number, and then, once the 
definition has been given, you think no more about 
it, because in reality it is not your definition that 
has taught you what a number is, you knew it long 
before, and when you come to write the word 
number farther on, you give it the same meaning 
as anybody else. In order to know what this 
meaning is, and if it is indeed the same in this 
phrase and in that, we must see how you have been 
led to speak of number and to introduce the word 
into the two phrases. I will not explain my point 
any further for the moment, for we shall have occa- 
sion to return to it. 

Thus we have a word to which we have explicitly 
given a definition A. We then proceed to make use 
of it in our text in a way which implicitly supposes 
another definition B. It is possible that these two 
definitions may designate the same object, but that 
such is the case is a new truth that must either be 
demonstrated or else admitted as an independent 

JVe shall see further on that the logicians have not 
fulfilled this second condition any better than the first. 



The definitions of number are very numerous and 
of great variety, and I will not attempt to enumerate 
even their names and their authors. We must not be 
surprised that there are so many. If any one of them 
was satisfactory we should not get any new ones. If 
each new philosopher who has applied himself to the 
question has thought it necessary to invent another, 
it is because he was not satisfied with those of his 
predecessors ; and if he was not satisfied, it was because 
he thought he detected 2. petitio principii. 

I have always experienced a profound sentiment 
of uneasiness in reading the works devoted to this 
problem. I constantly expect to run against a petitio 
principii, and when I do not detect it at once I am 
afraid that I have not looked sufficiently carefully. 

The fact is that it is impossible to give a definition 
without enunciating a phrase, and difficult to enun- 
ciate a phrase without putting in a name of number, 
or at least the word several, or at least a word in the 
plural. Then the slope becomes slippery, and every 
moment we are in danger of falling into the petitio 

I will concern myself in what follows with those 
only of these definitions in which the petitio principii 
is most skilfully concealed. 



The symbolical language created by Signor Peano 
plays a very large part in these new researches. It is 


capable of rendering some service, but it appears to 
me that M. Couturat attaches to it an exaggerated 
importance that must have astonished Peano himself. 

The essential element of this language consists in 
certain algebraical signs vi^hich represent the con- 
junctions : if, and, or, therefore. That these signs may 
be convenient is very possible, but that they should be 
destined to change the face of the whole philosophy is 
quite another matter. It is difficult to admit that 
the word if acquires, when written o, a virtue it did 
■iiot possess when written if. 

This invention of Peano was first called pasigraphy, 
that is to say, the art of writing a treatise on mathe- 
matics without using a single word of the ordinary 
language. This name defined its scope most exactly. 
Since then it has been elevated to a more exalted 
dignity, by having conferred upon it the title of 
logistic. The same word is used, it appears, in the Ecole 
de Guerre to designate the art of the quartermaster, 
the art of moving and quartering troops.* But no 
confusion need be feared, and we see at once that the 
new name implies the design of revolutionizing logic. 

We may see the new method at work in a mathe- 
matical treatise by Signor Burali-Forti entitled " Una 
Questione sui Numeri transfiniti" (An Enquiry concern- 
ing transfinite Numbers), included in Volume XI. of the 
" Rendiconti del circolo vtatematico di Palermo " (Reports 
of the mathematical club of Palermo). 

I will begin by saying that this treatise is very 
interesting, and, if I take it here as an example, it 

* In the French the confusion is with " lop'slique" the art of the 
"marechal des logis^'' or quartermaster. In English the possibility of 
confusion does not arise. 


is precisely because it is the most important of all 
that have been written in the new language. Besides, 
the uninitiated can read it, thanks to an interlined 
Italian translation. 

What gives importance to this treatise is the fact that 
it presented the first example of those antinomies met 
with in the study of transfinite numbers, which have 
become, during the last few years, the despair of 
mathematicians. The object of this note, says Signor 
Burali-Forti, is to show that there can be two trans- 
finite (ordinal) numbers, a and b, such that a is neither 
equal to, greater than, nor smaller than, b. 

The reader may set his mind at rest. In order to 
understand the considerations that will follow, he does 
not require to know what a transfinite ordinal number is. 

Now Cantor had definitely proved that between 
two transfinite numbers, as between two finite num- 
bers, there can be no relation other than equality or 
inequality in one direction or the other. But it is 
not of the matter of this treatise that I desire to speak 
here ; this would take me much too far from my 
subject. I only wish to concern myself with the form, 
and I ask definitely whether this form makes it gain 
much in the way of exactness, and whether it thereby 
compensates for the efforts it imposes upon the 
writer and the reader. 

To begin with, we find that Signor Burali-Forti 
defines the number i in the following manner : — 

I = t T' {Ko^(«,/^) e (ui One}, 

a definition eminently fitted to give an idea of the 
number i to people who had never heard it before. 
I do not understand Peanian well enough to ven- 


ture to risk a criticism, but I am very much afraid 
that this definition contains a petitio principii, seeing 
that I notice the figure i in the first half and the 
word One in the second. 

However that may be, Signer Burah-Forti starts 
with this definition, and, after a short calculation, 
arrives at the equation 

(27) I e No, 

which teaches us that One is a number. 

And since I am on the subject of these definitions 
of the first numbers, I may mention that M. Couturat 
has also defined both o and i. 

What is zero ? It is the number of elements in the 
class nil. And what is the class nil ? It is the class 
which contains none. 

To define zero as nil and nil as none is really an 
abuse of the wealth of language, and so M. Couturat 
has introduced an improvement into his definition by 

= 1 a: ^^ = A- 3- A = ixe<f>x), 

which means in English : zero is the number of the 
objects that satisfy a condition that is never fulfilled. 
But as never means zn no case, I do not see that any 
very great progress has been made. 

I hasten to add that the definition M. Couturat 
gives of the number i is more satisfactory. 

One, he says in substance, is the number of the 
elements of a class in which any two elements are 

It is more satisfactory, as I said, in this sense, 
that in order to define i, he does not use the word 
one ; on the other hand, he does use the word two. 


But I am afraid that if we asked M. Couturat what 
two is, he would be obliged to use the word one. 


But let us return to the treatise of Signor Burali- 
Forti. I said that his conclusions are in direct 
opposition to those of Cantor. Well, one day I 
received a visit from M. Hadamard, and the conversa- 
tion turned upon this antinomy. 

" Does not Burali-Forti's reasoning," I said, " seem 
to you irreproachable ? " 

" No," he answered ; " and, on the contrary, I have 
no fault to find with Cantor's. Besides, Burali-Forti 
had no right to speak of the whole of all the ordinal 

" Excuse me, he had that right, since he could 
always make the supposition that 

ft = T' (No, i >). 

I should like to know who could prevent him. And 
can we say that an object does not exist when we 
have called it 12 ? " 

It was quite useless ; I could not convince him 
(besides, ft would have been unfortunate if I had, since 
he was right). Was it only because I did not speak 
Feanian with sufficient eloquence ? Possibly, but, 
between ourselves, I do not think so. 

Thus, in spite of all this pasigraphical apparatus, 
the question is not solved, What does this prove? 
So long as it is merely a question of demonstrating 
that one is a number, pasigraphy is equal to the task ; 
but if a difficulty presents itself, if there is an anti- 
nomy to be resolved, pasigraphy becomes powerless. 




Russell's Logic. 

In order to justify its pretensions, logic has had to 
transform itself We have seen new logics spring 
up, and the most interesting of these is Mr. Bertrand 
Russell's. It seems as if there could be nothing new- 
written about formal logic, and as if Aristotle had gone 
to the very bottom of the subject. But the field 
that Mr. Russell assigns to logic is infinitely more 
extensive than that of the classical logic, and he 
has succeeded in expressing views on this subject that 
are original and sometimes true. 

To begin with, while Aristotle's logic was, above all, 
the logic of classes, and took as its starting-point 
the relation of subject and predicate, Mr. Russell 
subordinates the logic of classes to that of propositions. 
The classical syllogism, " Socrates is a man," etc., 
gives place to the hypothetical syllogism, " If A 
is true, B is true ; now if B is true, C is true, etc." 
This is, in my opinion, one of the happiest of ideas, 
for the classical syllogism is easily reduced to the 
hypothetical syllogism, while the inverse transforma- 
tion cannot be made without considerable difficulty. 


But this is not all. Mr. Russell's logic of propo- 
sitions is the study of the laws in accordance with 
which combinations are formed with the conjunctions 
if, and, or, and the negative not. This is a consider- 
able extension of the ancient logic. The properties of 
the classical syllogism can be extended without any 
difficulty to the hypothetical syllogism, and in the 
forms of this latter we can easily recognize the 
scholastic forms ; we recover what is essential in the 
classical logic. But the theory of the syllogism is still 
only the syntax of the conjunction if and, perhaps, 
of the negative. 

By adding two other conjunctions, and and or, 
Mr. Russell opens up a new domain to logic. The 
signs and and or follow the same laws as the two 
signs X and +, that is to say, the commutative, 
associative, and distributive laws. Thus and repre- 
sents logical multiplication, while or represents logical 
addition. This, again, is most interesting. 

Mr. Russell arrives at the conclusion that a false 
proposition of any kind involves all the other pro- 
positions, whether true or false. M. Couturat says 
that this conclusion will appear paradoxical at first 
sight. However, one has only to correct a bad 
mathematical paper to recognize how true Mr. 
Russell's view is. The candidate often takes an 
immense amount of trouble to find the first false 
equation ; but as soon as he has obtained it, it is 
no more than child's play for him to accumulate 
the most surprising results, some of which may 
actually be correct. 

(1.777) XI 



We see how much richer this new logic is than 
the classical logic. The symbols have been multiplied 
and admit of varied combinations, which are no longer 
of limited number. Have we any right to give this 
extension of meaning to the word logic} It would be 
idle to examine this question, and to quarrel with 
Mr. Russell merely on the score of words. We will 
grant him what he asks ; but we must not be sur- 
prised if we find that certain truths which had been 
declared to be irreducible to logic, in the old sense 
of the word, have become reducible to logic, in its 
new sense, which is quite different. 

We have introduced a large number of new notions, 
and they are not mere combinations of the old. 
Moreover, Mr. Russell is not deceived on this point, 
and not only at the beginning of his first chapter — that 
is to say, his logic of propositions — but at the beginning 
of his second and third chapters also — that is to say, 
his logic of classes and relations — he introduces new 
words which he declares to be undefinable. 

And that is not all. He similarly introduces prin- 
ciples which he declares to be undemonstrable. But 
these undemonstrable principles are appeals to in- 
tuition, a priori synthetic judgments. We regarded 
them as intuitive when we met them more or less 
explicitly enunciated in treatises on mathematics. 
Have they altered in character because the meaning 
of the word logic has been extended, and we find 
them now in a book entitled "Treatise on Logic"? 
They have not changed in nature, but only in position. 



Could these principles be considered as disguised 
definitions ? That they should be so, we should 
require to be able to demonstrate that they involve 
no contradiction. We should have to establish that, 
however far we pursue the series of deductions, we 
shall never be in danger of contradicting ourselves. 

We might attempt to argue as follows. We can 
verify the fact that the operations of the new logic, 
applied to premises free from contradiction, can only 
give consequences equally free from contradiction. If 
then, after n operations, we have not met with contra- 
diction, we shall not meet it any more after n+i. 
Accordingly, it is impossible that there can be a 
moment when contradiction will begin, which shows 
that we shall never meet it. Have we the right 
to argue in this way? No, for it would be making 
complete induction, and we must not forget that 
we do not yet know the principle of complete induction. 

Therefore we have no right to regard these axioms 
as disguised definitions, and we have only one course 
left. Each one of them, we admit, is a new act of 
intuition. This is, moreover, as I believe, the thought 
of Mr. Russell and M. Couturat. 

Thus each of the nine undefinable notions and 
twenty undemonstrable propositions (I feel sure that, 
if I had made the count, I should have found one 
or two more) which form the groundwork of the 
new logic — of the logic in the broad sense — pre- 
supposes a new and independent act of our intuition, 
and why should we not term it a true a /r^Wz synthetic 
judgment.? On this point everybody seems to be 


agreed ; but what Mr. Russell claims, and what appears 
to me doubtful, is that after these appeals to intuition 
we shall have finished : we shall have no more to make, 
and we shall be able to construct the whole of mathe- 
matics without bringing in a single new element. 


M. Couturat is fond of repeating that this new logic 
is quite independent of the idea of number. I will 
not amuse myself by counting how many instances 
his statement contains of adjectives of number, 
cardinal as well as ordinal, or of indefinite adjectives 
such as several. However, I will quote a few 
examples : — 

" The logical product of two or of several propo- 
sitions is " 

" All propositions are susceptible of two values only, 
truth or falsehood." 

" The relative product of two relations is a relation." 
" A relation is established between two terms." 
Sometimes this difficulty would not be impossible 
to avoid, but sometimes it is essential. A relation is 
incomprehensible without two terms. It is impossible 
to have the intuition of a relation, without having 
at the same time the intuition of its two terms, and 
without remarking that they are two, since, for a 
relation to be conceivable, they must be two and 
two only. 



I come now to what M. Couturat calls the ordinal 
theory, which is the groundwork of arithmetic properly 


so called. M. Couturat begins by enunciating Peano's 
five axioms, which are independent, as Signor Peano 
and Signor Padoa have demonstrated. 

1. Zero is a whole number. 

2. Zero is not the sequent of any whole number. 

3. The sequent of a whole number is a whole 
number. To which it would be well to add : every 
whole number has a sequent. 

4. Two whole numbers are equal if their sequents 
are equal. 

The Sth axiom is the principle ot complete induction. 

M. Couturat considers these axioms as disguised 
definitions ; they constitute the definition by postulates 
of zero, of the " sequent," and of the whole number. 

But we have seen that, in order to allow of a 
definition by postulates being accepted, we must be 
able to establish that it implies no contradiction. 

Is this the case here ? Not in the very least. 

The demonstration cannot be made by example. 
We cannot select a portion of whole numbers — for 
instance, the three first — and demonstrate that they 
satisfy the definition. 

If I take the series o, r, 2, I can readily see that 
it satisfies axioms i, 2, 4, and 5 ; but in order that 
it should satisfy axiom 3, it is further necessary that 
3 should be a whole number, and consequently that 
the series o, I, 2, 3 should satisfy the axioms. We 
could verify that it satisfies axioms I, 2, 4, and 5, 
but axiom 3 requires besides that 4 should be a 
whole number, and that the series o, i, 2, 3, 4 should 
satisfy the axioms, and so on indefinitely. 

It is, therefore, impossible to demonstrate the 
axioms for some whole numbers without demonstrat- 


ing them for all, and so we must give up the 
demonstration by example. 

It is necessary, then, to take all the consequences 
of our axioms and see whether they contain any 
contradiction. If the number of these consequences 
were finite, this would be easy ; but their number 
is infinite — they are the whole of mathematics, or at 
least the whole of arithmetic. 

What are we to do, then ? Perhaps, if driven to 
it, we might repeat the reasoning of Section III. 
But, as I have said, this reasoning is complete induction, 
and it is precisely the principle of complete induction 
that we are engaged in justifying. 


Hilbert's Logic. 

I come now to Mr. Hilbert's important work, 
addressed to the Mathematical Congress at Heidelberg, 
a French translation of which, by M. Pierre Boutroux, 
appeared in I' Enseignement Mathematique, while an 
English translation by Mr. Halsted appeared in The 
Monist. In this work, in which we find the most 
profound thought, the author pursues an aim similar 
to Mr. Russell's, but he diverges on many points from 
his predecessor. 

" However,'' he says, " if we look closely, we recog- 
nize that in logical principles, as they are com- 
monly presented, certain arithmetical notions are 
found already implied ; for instance, the notion of 
whole, and, to a certain extent, the notion of number. 
Thus we find ourselves caught in a circle, and that 
is why it seems to me necessary, if we wish to avoid 


all paradox, to develop the principles of logic and of 
arithmetic simultaneously." 

We have seen above that what Mr. Hilbert says 
of the principles of logic, as they are commonly pre- 
sented, applies equally to Mr. Russell's logic. For 
Mr. Russell logic is anterior to arithmetic, and for 
Mr. Hilbert they are " simultaneous." Further on we 
shall find other and yet deeper differences ; but we 
will note them as they occur. I prefer to follow the 
development of Hilbert's thought step by step, quoting 
the more important passages verbatim. 

" Let us first take into consideration the object i." 
We notice that in acting thus we do not in any way 
imply the notion of number, for it is clearly understood 
that I here is nothing but a symbol, and that we do 
not in any way concern ourselves with knowing its 
signification. " The groups formed with this object, 
two, three, or several times repeated . . ." This 
time the case is quite altered, for if we introduce the 
words two, three, and, above all, several, we introduce 
the notion of number ; and then the definition of the 
finite whole number that we find later on comes a 
trifle late. The author was much too wary not to 
perceive this petitio principii. And so, at the end of 
his work, he seeks to effect a real patching-up. 

Hilbert then introduces two simple objects, i and 
=, and pictures all the combinations of these two 
objects, all the combinations of their combinations, 
and so on. It goes without saying that we must 
forget the ordinary signification of these two signs, 
and not attribute any to them. He then divides these 
combinations into two classes, that of entities and that 
of nonentities, and, until further orders, this partition 


is entirely arbitrary. Every affirmative proposition 
teaches us that a combination belongs to the class of 
entities, and every negative proposition teaches us 
that a certain combination belongs to the class ot 


We must now note a difference that is of the 
highest importance. For Mr. Russell a chance object, 
which he designates by ;r, is an absolutely indeterminate 
object, about which he assumes nothing- For Hilbert 
it is one of those combinations formed with the symbols 
I and = ; he will not allow the introduction of any- 
thing but combinations of objects already defined. 
Moreover, Hilbert formulates his thought in the most 
concise manner, and I think I ought to reproduce 
his statement in extenso : " The indeterminates which 
figure in the axioms (in place of the 'some' or the 
' all ' of ordinary logic) represent exclusively the whole 
of the objects and combinations that we have already 
acquired in the actual state of the theory, or that we 
are in course of introducing. Therefore, when we 
deduce propositions from the axioms under considera- 
tion, it is these objects and these combinations alone 
that we have the right to substitute for the indeter- 
minates. Neither must we forget that when we 
increase the number of the fundamental objects, the 
axioms at the same time acquire a new extension, and 
must, in consequence, be put to the proof afresh and, 
if necessary, modified." 

The contrast with Mr. Russell's point of view is 
complete. According to this latter philosopher, we 
may substitute in place of x not only objects already 


known, but anything whatsoever. Russell is faithful 
to his point of view, which is that of comprehension. 
He starts with the general idea of entity, and enriches 
it more and more, even while he restricts it, by adding 
to it new qualities. Hilbert, on the contrary, only 
recognizes as possible entities combinations of objects 
already known ; so that (looking only at one side of 
his thought) we might say that he takes the point 
of view of extension. 


Let us proceed with the exposition of Hilbert's 
ideas. He introduces two axioms which he enunciates 
in his symbolical language, but which signify, in the 
language of the uninitiated like us, that every quantity 
is equal to itself, and that every operation upon two 
identical quantities gives identical results. So stated 
they are evident, but such a presentation of them 
does not faithfully represent Hilbert's thought. For 
him mathematics has to combine only pure symbols, 
and a true mathematician must base his reasoning 
upon them without concerning himself with their 
meaning. Accordingly, his axioms are not for him 
what they are for the ordinary man. 

He considers them as representing the definition by 
postulates of the symbol :=, up to this time devoid 
of all signification. But in order to justify this defini- 
tion, it is necessary to show that these two axioms do 
not lead to any contradiction. 

For this purpose Hilbert makes use of the reasoning 
of Section HL, without apparently perceiving that he 
is making complete induction. 



The end of Mr. Hilbert's treatise is altogether 
enigmatical, and I will not dwell upon it. It is full 
of contradictions, and one feels that the author is 
vaguely conscious of the petitio principii he has been 
guilty of, and that he is vainly trying to plaster up 
the cracks in his reasoning. 

What does this mean ? It means that when he 
co77ies to demonstrate that the definition of the whole 
number by the axiom of complete induction does not 
involve contradiction, Mr. Hilbert breaks down, just as 
Mr. Russell and M. Couturat broke down, because the 
difficulty is too great. 



Geometry, M. Couturat says, is a vast body of 
doctrine upon which complete induction does not 
intrude. This is true to a certain extent : we cannot 
say that it does not intrude at all, but that it intrudes 
very little. If we refer to Mr. Halsted's " Rational 
Geometry " (New York : John Wiley and Sons, 
1904), founded on Hilbert's principles, we find the 
principle of induction intruding for the first time 
at page 114 (unless, indeed, 1 have not searched care- 
fully enough, which is quite possible). 

Thus geometry, which seemed, only a few years 
ago, the domain in which intuition held undisputed 
sway, is to-day the field in which the logisticians 
appear to triumph. Nothing could give a better 
measure of the importance of Hilbert's geometrical 
works, and of the profound impression they have left 
upon our conceptions. 


But we must not deceive ourselves. What is, in 
fad, the fundamental theorem of geometry ? It is that 
the axioms of geometry do not involve contradiction, and 
this cannot be demonstrated without the principle of 

How does Hilbert demonstrate this essential point ? 
He does it by relying upon analysis, and, through it, 
upon arithmetic, and, through it, upon the principle 
of induction. 

If another demonstration is ever discovered, it will 
still be necessary to rely on this principle, since the 
number of the possible consequences of the axioms 
which we have to show are not contradictory is 



Our conclusion is, first of all, that the principle of 
induction cannot be regarded as the disguised definition 
of the whole number. 

Here are three truths : — 

The principle of complete induction ; 
Euclid's postulate ; 

The physical law by which phosphorus melts 
at 44° centigrade (quoted by M. Le Roy). 

We say : these are three disguised definitions — the 
first that of the whole number, the second that of the 
straight line, and the third that of phosphorus. 

I admit it for the second, but I do not admit it 
for the two others, and I must explain the reason of 
this apparent inconsistency. 

In the first place, we have seen that a definition 


is only acceptable if it is established that it does not 
involve contradiction. We have also shown that, in 
the case of the first definition, this demonstration is 
impossible ; while in the case of the second, on the 
contrary, we have just recalled the fact that Hilbert 
has given a complete demonstration. 

So far as the third is concerned, it is clear that it 
does not involve contradiction. But does this mean 
that this definition guarantees, as it should, the 
existence of the object defined ? We are here no 
longer concerned with the mathematical sciences, but 
with the physical sciences, and the word existence has 
no longer the same meaning ; it no longer signifies 
absence of contradiction, but objective existence. 

This is one reason already for the distinction I make 
between the three cases, but there is a second. In 
the applications we have to make of these three 
notions, do they present themselves as defined by 
these three postulates ? 

The possible applications of the principle of induc- 
tion are innumerable. Take, for instance, one of those 
we have expounded above, in which it is sought to 
establish that a collection of axioms cannot lead to 
a contradiction. For this purpose we consider one of 
the series of syllogisms that can be followed out, start- 
ing with these axioms as premises. 

When we have completed the n*'^ syllogism, we see 
that we can form still another, which will be the 
(«+i)''^: thus the number n serves for counting a 
series of successive operations ; it is a number that 
can be obtained by successive additions. Accordingly, 
it is a number from which we can return to unity by 
successive subtractions. It is evident that we could 


not do so if we had n = n-i, for then subtraction 
would always give us the same number. Thus, then, 
the way in which we have been brought to consider 
this number n involves a definition of the finite whole 
number, and this definition is as follows: a finite 
whole number is that which can be obtained by suc- 
cessive additions, and which is such that n is not equal 
to n-l. 

This being established, what do we proceed to do ? 
We show that if no contradiction has occurred up to 
the w'* syllogism, it will not occur any the more at 
the {n+iy^, and we conclude that it will never occur. 
You say I have the right to conclude thus, because 
whole numbers are, by definition, those for which such 
reasoning is legitimate. But that involves another 
definition of the whole number, which is as follows : 
a whole numbej- is that about which we can reason by 
recurrence. In the species it is that of which we can 
state that, if absence of contradiction at the moment 
of occurrence of a syllogism whose number is a whole 
number carries with it the absence of contradiction 
at the moment of occurrence of the syllogism whose 
number is the following whole number, then we need 
not fear any contradiction for any of the syllogisms 
whose numbers are whole numbers. 

The two definitions are not identical. They are 
equivalent, no doubt, but they are so by virtue of an 
a priori synthetic judgment; we cannot pass from 
one to the other by purely logical processes. Con- 
sequently, we have no right to adopt the second after 
having introduced the whole number by a road which 
presupposes the first. 

On the contrary, what happens in the case of the 


straight line ? I have already explained this so often 
that I feel some hesitation about repeating myself 
once more. I will content myself with a brief sum- 
mary of my thought. 

We have not, as in the previous case, two equivalent 
definitions logically irreducible one to the other. We 
have only one expressible in words. It may be said 
that there is another that we feel without being able 
to enunciate it, because we have the intuition of a 
straight line, or because we can picture a straight 
line. But, in the first place, we cannot picture it in 
geometric space, but only in representative space ; 
and then we can equally well picture objects which 
possess the other properties of a straight line, and 
not that of satisfying Euclid's postulate. These 
objects are "non- Euclidian straight lines," which, 
from a certain point of view, are not entities 
destitute of meaning, but circles (true circles of true 
space) orthogonal to a certain sphere. If, among 
these objects equally susceptible of being pictured, 
it is the former (the Euclidian straight lines) that 
we call straight lines, and not the latter (the non- 
Euclidian straight lines), it is certainly so by definition. 

And if we come at last to the third example, the 
definition of phosphorus, we see that the true defini- 
tion would be: phosphorus is this piece of matter 
that I see before me in this bottle. 


Since I am on the subject, let me say one word 
more. Concerning the example of phosphorus, I 
said : " This proposition is a true physical law that 
can be verified, for it means : all bodies which possess 


all the properties of phosphorus except its melting- 
point, melt, as it does, at 44° centigrade." It has been 
objected that this law is not verifiable, for if we came 
to verify that two bodies resembling phosphorus melt 
one at 44° and the other at 50° centigrade, we could 
always say that there is, no doubt, besides the melting- 
point, some other property in which they differ. 

This was not exactly what I meant to say, and I 
should have written : " all bodies which possess such 
and such properties in finite number (namely, the 
properties of phosphorus given in chemistry books, 
with the exception of its melting-point) melt at 44° 

In order to make still clearer the difference between 
the case of the straight line and that of phosphorus, 
I will make one more remark. The straight line has 
several more or less imperfect images in nature, the 
chief of which are rays of light and the axis of 
rotation of a solid body. Assuming that we ascertain 
that the ray of light does not satisfy Euclid's postulate 
(by showing, for instance, that a star has a negative 
parallax), what shall we do ? Shall we conclude that, 
as a straight line is by definition the trajectory of 
light, it does not satisfy the definition, or, on the 
contrary, that, as a straight line by definition satisfies 
the postulate, the ray of light is not rectilineal ? 

Certainly we are free to adopt either definition, 
and, consequently, either conclusion. But it would be 
foolish to adopt the former, because the ray of light 
probably satisfies in a most imperfect way not only 
Euclid's postulate but the other properties of the 
straight line ; because, while it deviates from the 
Euclidian straight, it deviates none the less from the 


axis of rotation of solid bodies, which is another 
imperfect image of the, straight line ; and lastly, 
because it is, no doubt, subject to change, so that 
such and such a line which was straight yesterday- 
will no longer be so to-morrow if some physical cir- 
cumstance has altered. 

Assume, now, that we succeed in discovering that 
phosphorus melts not at 44° but at 43'9° centigrade. 
Shall we conclude that, as phosphorus is by definition 
that which melts at 44°, this substance that we called 
phosphorus is not true phosphorus, or, on the contrary, 
that phosphorus melts at 43 '9°? Here, again, we are 
free to adopt either definition, and, consequently, either 
conclusion ; but it would be foolish to adopt the 
former, because we cannot change the name of a 
substance every time we add a fresh decimal to its 


To sum up, Mr. Russell and Mr. Hilbert have both 
made a great effort, and have both of them written 
a book full of views that are original, profound, and 
often very true. These two books furnish us with 
subject for much thought, and there is much that we 
can learn from them. Not a few of their results are 
substantial and destined to survive. 

But to say that they have definitely settled the 
controversy between Kant and Leibnitz and destroyed 
the Kantian theory of mathematics is evidently un- 
true. I do not know whether they actually imagined 
they had done it, but if they did they were mistaken. 




The logisticians have attempted to answer the fore- 
going considerations. For this purpose they have 
been obHged to transform logistic, and Mr. Russell 
in particular has modified his original views on certain 
points. Without entering into the details of the con- 
troversy, I should like to return to what are, in my 
opinion, the two most important questions. Have the 
rules of logistic given any proof of fruitfulness and of 
infallibility? Is it true that they make it possible to 
demonstrate the principle of complete induction with- 
out any appeal to intuition ? 


The Infallibility of Logistic. 

As regards fruitfulness, it seems that M. Couturat 
has most childish illusions. Logistic, according to 
him, lends " stilts and wings " to discovery, and on the 
following page he says, " // is ten years since Signor 
Peano published the first edition of his " Formulaire." 

What ! you have had wings for ten years, and you 
haven't flown yet ! 

I have the greatest esteem for Signor Peano, who 

(1.777) 12 


has done some very fine things (for instance, his curve 
which fills a whole area) ; but, after all, he has not 
gone any farther, or higher, or faster than the majority 
of wingless mathematicians, and he could have done 
everything just as well on his feet. 

On the contrary, I find nothing in logistic for the 
discoverer but shackles. It does not help us at all 
in the direction of conciseness, far from it ; and if it 
requires 27 equations to establish that I is a num- 
ber, how many will it require to demonstrate a real 
theorem ? If we distinguish, as Mr. Whitehead does, 
the individual x, the class whose only member is x, 
which we call ix, then the class whose only member 
is the class whose only member is x, which we call 
ux, do we imagine that these distinctions, however 
useful they may be, will greatly expedite our progress ? 

Logistic forces us to say all that we commonly 
assume, it forces us to advance step by step ; it is 
perhaps surer, but it is not more expeditious. 

It is not wings you have given us, but leading- 
strings. But we have the right to demand that these 
leading-strings should keep us from falling ; this is 
their only excuse. When an investment does not pay 
a high rate of interest, it must at least be a gilt-edged 

Must we follow your rules blindly? Certainly, for 
otherwise it would be intuition alone that would enable 
us to distinguish between them. But in that case they 
must be infallible, for it is only in an infallible author- 
ity that we can have blind confidence. Accordingly, 
this is a necessity for you : you must be infallible or 
cease to exist. 

You have no right to say to us: " We make mistakes, 


it is true, but you make mistakes too.'' For us, making 
mistakes is a misfortune, a very great misfortune, but 
for you it is death. 

Neither must you say, " Does the infallibility of arith- 
metic prevent errors of addition ? " The rules of calcula- 
tion are infallible, and yet we find people making 
mistakes through not applying these rules. But a 
revision of their calculation will show at once just 
where they went astray. Here the case is quite dif- 
ferent. The logisticians have applied their rules, and 
yet they have fallen into contradiction. So true is 
this, that they are preparing to alter these rules and 
•'sacrifice the notion of class." Why alter them if 
they were infallible ? 

" We are not obliged," you say, " to solve hie et nunc 
all possible problems." Oh, we do not ask as much as 
that. If, in face of a problem, you gave no solution, 
we should have nothing to say ; but, on the contrary, 
you give two, and these two are contradictory, and 
consequently one at least of them is false, and it is 
this that constitutes a failure. 

Mr. Russell attempts to reconcile these contradic- 
tions, which can only be done, according to him, " by 
restricting or even sacrificing the notion of class." 
And M. Couturat, discounting the success of this 
attempt, adds: "If logisticians succeed where others 
have failed, M. Poincare will surely recollect this sen- 
tence, and give logistic the credit of the solution." 

Certainly not Logistic exists ; it has its code, which 
has already gone through four editions ; or, rather, it 
is this code which is logistic itself Is Mr. Russell 
preparing to show that one at least of the two contra- 
dictory arguments has transgressed the code ? Not in 


the very least ; he is preparing to alter these laws and 
to revoke a certain number of them. If he succeeds, 
I shall give credit to Mr. Russell's intuition, and not to 
Peanian Logistic, which he will have destroyed. 

Liberty of Contradiction. 

I offered two principal objections to the definition 
of the whole number adopted by the logisticians. 
What is M. Couturat's answer to the first of these 
objections ? 

What is the meaning in mathematics of the word 
to exist? It means, I said, to be free from contradic- 
tion. This is what M. Couturat disputes. " Logical 
existence," he says, " is quite a different thing from 
absence of contradiction. It consists in the fact that 
a class is not empty. To say that some «'s exist is, 
by definition, to assert that the class a is not void." 
And, no doubt, to assert that the class a is not void 
is, by definition, to assert that some ds exist. But 
one of these assertions is just as destitute of meaning 
as the other if they do not both signify either that 
we can see or touch a, which is the meaning given 
them by physicists or naturalists, or else that we can 
conceive of an a without being involved in contradic- 
tions, which is the meaning given them by logicians 
and mathematicians. 

In M. Couturat's opinion it is not non-contradiction 
that proves existence, but existence that proves non- 
contradiction. In order to establish the existence of a 
class, we must accordingly establish, by an example, 
that there is an individual belonging to that class. 


"But it will be said, How do we demonstrate the 
existence of this individual ? Is it not necessary that 
this existence should be established, to enable us to 
deduce the existence of the class of which it forms 
part? It is not so. Paradoxical as the assertion 
may appear, we never demonstrate the existence of 
an individual. Individuals, from the very fact that 
they are individuals, are always considered as existing. 
We have never to declare that an individual exists, 
absolutely speaking, but only that it exists in a class." 
M. Couturat finds his own assertion paradoxical, and 
he will certainly not be alone in so finding it. Never- 
theless it must have some sense, and it means, no 
doubt, that the existence of an individual alone in 
the world, of which nothing is asserted, cannot involve 
contradiction. As long as it is quite alone, it is 
evident that it cannot interfere with any one. Well, 
be it so ; we will admit the existence of the individual, 
" absolutely speaking," but with it we have nothing to 
do. It still remains to demonstrate the existence 
of the individual " in a class," and, in order to do 
this, you will still have to prove that the assertion that 
such an individual belongs to such a class is neither 
contradictory in itself nor with the other postulates 

" Accordingly," M. Couturat continues, " to assert 
that a definition is not valid unless it is first proved 
that it is not contradictory, is to impose an arbitrary 
and improper condition." The claim for the liberty 
of contradiction could not be stated in more emphatic 
or haughtier terms. " In any case, the onus probandi 
rests with those who think these principles are contra- 
dictory." Postulates are presumed to be compatible. 


just as a prisoner is presumed to be innocent, until the 
contrary is proved. 

It is unnecessary to add that I do not acquiesce 
in this claim. But, you say, the de-honstration you 
demand of us is impossible, and yoa cannot require 
us to " aim at the moon." Excuse me; it is impossible 
for you, but not for us who admit the principle of 
induction as an a priori synthetic judgment. This 
would be necessary for you as it is for us. 

In order to demonstrate that a system of postulates 
does not involve contradiction, it is necessary to apply 
the principle of complete induction. Not only is there 
nothing " extraordinary " in this method of reasoning, 
but it is the only correct one. It is not " incon- 
ceivable " that any one should ever have used it, and 
it is not difficult to find "examples and precedents." 
In my article I have quoted two, and they were 
borrowed from Hilbert's pamphlet. He is not alone 
in having made use of it, and those who have 
not done so have been wrong. What I reproach 
Hilbert with, is not that he has had recourse to it 
(a born mathematician such as he could not but see 
that a demonstration is required, and that this is the 
only possible one), but that he has had recourse to it 
without recognizing the reasoning by recurrence. 


The Second Objection. 

1 had noted a second error of the logisticians in 
Hilbert's article. To-day Hilbert is excommuni- 
cated, and M. Couturat no longer considers him as 
a logistician. He will therefore, ask me if I have 


found the same mistake in the orthodox logfis- 
ticians. I have not seen it in the pages I have read, 
but I do not know vi^hether I should find it in the 
three hundred pages they have written that I have no 
wish to read. 

Only, they will have to commit the error as soon 
as they attempt to make any sort of an application 
of mathematical science. The eternal contemplation 
of its own navel is not the sole object of this science. 
It touches nature, and one day or other it will come 
into contact with it. Then it will be necessary to 
shake off purely verbal definitions and no longer to 
content ourselves with words. 

Let us return to Mr. Hilbert's example. It is still 
a question of reasoning by recurrence and of knowing 
whether a system of postulates is not contradictory. 
M. Couturat will no doubt tell me that in that case 
it does not concern him, but it may perhaps interest 
those who do not claim, as he does, the liberty of 

We wish to establish, as above, that we shall not 
meet with contradiction after some particular number 
of arguments, a number which may be as large as you 
please, provided it is finite. For this purpose we 
must apply the principle of induction. Are we to 
understand here by finite number every number to 
which the principle of induction applies ? Evidently 
not, for otherwise we should be involved in the most 
awkward consequences. 

To have the right to lay down a system of postu- 
lates, we must be assured that they are not contra- 
dictory. This is a truth that is admitted by the 
majority of scientists ; I should have said all before 


reading M. Couturat's last article. But what does it 
signify ? Does it mean that we must be sure of not 
meeting with contradiction after a finite number of 
propositions, the finite number being, by definition, 
that which possesses all the properties of a recurrent 
nature in such a way that if one of these properties 
were found wanting — if, for instance, we came upon a 
contradiction — we should agree to say that the number 
in question was not finite ? 

In other words, do we mean that we must be sure 
of not meeting a contradiction, with this condition, 
that we agree to stop just at the moment when we are 
on the point of meeting one ? The mere statement 
of such a proposition is its sufficient condemnation. 

Thus not only does Mr. Hilbert's reasoning assume 
the principle of induction, but he assumes that this 
principle is given us, not as a simple definition, but 
as an a priori synthetic judgment. 

I would sum up as follows : — 

A demonstration is necessary. 

The only possible demonstration is the demonstra- 
tion by recurrence. 

This demonstration is legitimate only if the prin- 
ciple of induction is admitted, and if it is regarded 
not as a definition but as a synthetic judgment. 


The Cantorian Antinomies. 

I will now take up the examination of Mr. Russell's 
new treatise. This treatise was written with the object 
ofovercoming the difficulties raised by those Cantorian 


antinomies to which I have already made frequent 
allusion. Cantor thought it possible to construct a 
Science of the Infinite. Others have advanced further 
along the path he had opened, but they very soon ran 
against strange contradictions. These antinomies are 
already numerous, but the most celebrated are : — 

1. Burali-Forti's antinomy. 

2. The Zermelo-Konig antinomy. 

3. Richard's antinomy. 

Cantor had demonstrated that ordinal numbers (it 
is a question of transfinite ordinal numbers, a new 
notion introduced by him) can be arranged in a lineal 
series ; that is to say, that of two unequal ordinal 
numbers, there is always one that is smaller than the 
other. Burali-Forti demonstrates the contrary ; and 
indeed, as he says in substance, if we could arrange all 
the ordinal numbers in a lineal series, this series 
would define an ordinal number that would be 
greater than all the others, to which we could then 
add I and so obtain yet another ordinal number 
which would be still greater. And this is contra- 

We will return later to the Zermelo-Konig anti- 
nomy, which is of a somewhat different nature. 
Richard's antinomy is as follows {Revue ginerale des 
Sciences, June 30, 1905). Let us consider all the 
decimal numbers that can be defined with the help of 
a finite number of words. These decimal numbers form 
an aggregate E, and it is easy to see that this aggregate 
is denumerable — that is to say, that it is possible to 
number the decimal numbers of this aggregate from one 
to infinity. Suppose the numeration effected, and let 


us define a number N in the following manner. If 
the «'" decimal of the «'" number of the aggregate E is 

o, I, 2, 3, 4, 5, 6, 7, 8, or 9, 
the n'^ decimal of N will be 

r, 2, 3, 4, 5, 6, 7, 8, I, or i. 

As we see, N is not equal to the n'" number of E, 
and since n is any chance number, N does not belong 
to E, and yet N should belong to this aggregate, since 
we have defined it in a finite number of words. 

We shall see further on that M. Richard himself 
has, with much acuteness, given the explanation of his 
paradox, and that his explanation can be extended, 
mutatis mutandis, to the other paradoxes of like 
nature. Mr. Russell quotes another rather amusing 
antinomy : 

What is the smallest whole number that cannot be 
defined in a sentence formed of less than a hundred 
English words ? 

This number exists, and, indeed, the number of 
numbers capable of being defined by such a sentence 
is evidently finite, since the number of words in the 
English language is not infinite. Therefore among 
them there will be one that is smaller than all the 

On the other hand the number does not exist, for 
its definition involves contradiction. The number, in 
fact, is found to be defined by the sentence in italics, 
which is formed of less than a hundred English words, 
and, by definition, the number must not be capable 
of being defined by such a sentence. 


Zigzag Theory and No Classes Theory. 

What is Mr. Rus.sell's attitude in face of these con- 
tradictions ? After analysing those I have just spoken 
of, and quoting others, after putting them in a form 
that recalls Epimenides, he does not hesitate to con- 
clude as follows : — 

" A propositional function of one variable does not 
always determine a class." * A " propositional func- 
tion " (that is to say, a definition) or " norm " can be 
"non-predicative." And this does not mean that these 
non-predicative propositions determine a class that is 
empty or void ; it does not mean that there is no 
value oi X that satisfies the definition and can be one of 
the elements of the class. The elements exist, but they 
have no right to be grouped together to form a class. 

But this is only the beginning, and we must know 
how to recognize whether a definition is or is not 
predicative. For the purpose of solving this problem, 
Mr. Russell hesitates between three theories, which he 
calls — 

A. The zigzag theory. 

B. The theory of limitation of size. 

C. The no classes theory. 

According to the zigzag theory, "definitions (pro- 
positional functions) determine a class when they are 
fairly simple, and only fail to do so when they are 
complicated and recondite." Now who is to decide 

* This and the following quotations are from Mr. Russell's paper, 
" On some difficulties in the theory of transPnite numbers and order 
types, ^'Proceedings of the London Mathematical Society. Ser. 2, Vol. 4, 
Part I. 


whether a definition can be regarded as sufficiently 
simple to be acceptable ? To this question we get no 
answer except a candid confession of powerlessness. 
" The axionns as to what functions are predicative 
have to be exceedingly complicated, and cannot be 
recommended by any intrinsic plausibility. This is a 
defect which might be remedied by greater ingenuity, 
or by the help of some hitherto unnoticed distinction. 
But hitherto, in attempting to set up axioms for this 
theory, I have found no guiding principle except the 
avoidance of contradictions." 

This theory therefore remains very obscure. In the 
darkness there is a single glimmer, and that is the 
word zigzag. What Mr. Russell calls zigzagginess is 
no doubt this special character which distinguishes the 
argument of Epimenides. 

According to the theory of limitation of size, a 
class must not be too extensive. It may, perhaps, 
be infinite, but it must not be too infinite. 

But we still come to the same difficulty. At what 
precise moment will it begin to be too extensive ? Of 
course this difficulty is not solved, and Mr. Russell 
passes to the third theory. 

In the no classes theory all mention of the word 
class is prohibited, and the word has to be replaced by 
various periphrases. What a change for the logis- 
ticians who speak of nothing but class and classes of 
classes ! The whole of Logistic will have to be re- 
fashioned. Can we imagine the appearance of a page 
of Logistic when all propositions dealing with class 
have been suppressed ? There will be nothing left 
but a few scattered survivors in the midst of a blank 
page. Apparent rari nantes in gurgite vasto. 


However that may be, we understand Mr. Russell's 
hesitation at the modifications to which he is about 
to submit the fundamental principles he has hitherto 
adopted. Criteria will be necessary to decide whether 
a definition is too complicated or too extensive, and 
these criteria cannot be justified except by an appeal 
to intuition. 

It is towards the no classes theory that Mr. Russell 
eventually inclines. 

However it be, Logistic must be refashioned, and it 
is not yet known how much of it can be saved. It is 
unnecessary to add that it is Cantorism and Logistic 
alone that are in question. The true mathematics, the 
mathematics that is of some use, may continue to 
develop according to its own principles, taking no 
heed of the tempests that rage without, and step 
by step it will pursue its wonted conquests, which are 
decisive and have never to be abandoned. 

The True Solution. 

How are we to choose between these different 
theories? It seems to me that the solution is con- 
tained in M. Richard's letter mentioned above, which 
will be found in the Revue G'enerale des Sciences of June 
30, 1905. After stating the antinomy that I have called 
Richard's antinomy, he gives the explanation. 

Let us refer to what was said of this antinomy in 
Section V. E is the aggregate oi all the numbers that 
can be defined by a finite number of words, without 
introducing the notion of the aggregate E itself, otherwise 


the definition of E would contain a vicious circle, for 
we cannot define E by the aggregate E itself. 

Now we have defined N by a finite number of 
words, it is true, but only with the help of the notion 
of the aggregate E, and that is the reason why N does 
not form a part of E. 

In the example chosen by M. Richard, the con- 
clusion is presented with complete evidence, and the 
evidence becomes the more apparent on a reference to 
the actual text of the letter. But the same explana- 
tion serves for the other antinomies, as may be easily 

Thus the definitions that must be regarded as non- 
predicative are those which contain a vicious circle. 
The above examples show sufficiently clearly what 
I mean by this. Is this what Mr. Russell calls 
" zigzagginess " ? I merely ask the question without 
answering it. 


The Demonstrations of the Principle 
OF Induction. 

We will now examine the so-called demonstrations 
of the principle of induction, and more particularly 
those of Mr. Whitehead and Signor Burali-Forti. 

And first we will speak of Whitehead's, availing our- 
selves of some new denominations happily introduced 
by Mr. Russell in his recent treatise. 

We will call recurrent class every class of numbers 
that includes zero, and also includes « + 1 if it 
includes ;/. 

We will call inductive number every number which 
forms a part of all recurrent classes. 


Upon what condition will this latter definition, 
which plays an essential part in Whitehead's demon- 
stration, be " predicative" and consequently acceptable? 

Following upon what has been said above, we must 
understand by all recurrent classes all those whose 
definition does not contain the notion of inductive 
number ; otherwise we shall be involved in the vicious 
circle which engendered the antinomies. 

Now, Whitehead has not taken this precaution. 

Whitehead's argument is therefore vicious ; it is the 
same that led to the antinomies. It was illegitimate 
when it gave untrue results, and it remains illegitimate 
when it leads by chance to a true result. 

A definition which contains a vicious circle defines 
nothing. It is of no use to say we are sure, whatever 
be the meaning given to our definition, that there is 
at least zero which belongs to the class of inductive 
numbers. It is not a question of knowing whether 
this class is empty, but whether it can be rigidly 
delimited. A " non-predicative class " is not an empty 
class, but a class with uncertain boundaries. 

It is unnecessary to add that this particular objection 
does not invalidate the general objections that apply 
to all the demonstrations. 


Signor Burali-Forti has given another demonstration 
in his article " Le Classi finite" {Atti di Torino, 
Vol. xxxii). But he is obliged to admit two postulates : 

The first is that there exists always at least one 
infinite class. 

The second is stated thus : — 

« e K (K - t /\). 3. 2< < v u. 


The first postulate is no more evident than the 
principle to be demonstrated. The second is not 
only not evident, but it is untrue, as Mr. Whitehead 
has shown, as, moreover, the veriest schoolboy could 
have seen at the first glance if the axiom had been 
stated in intelligible language, since it means : the 
number of combinations that can be formed with 
several objects is smaller than the number of those 


Zermelo's Axiom. 

In a celebrated demonstration. Signer Zermelo 
relies on the following axiom : 

In an aggregate of any kind (or even in each of 
the aggregates of an aggregate of aggregates) we 
can always select one element at random (even if 
the aggregate of aggregates contains an infinity 
of aggregates). 

This axiom had been applied a thousand times with- 
out being stated, but as soon as it was stated, it raised 
doubts. Some mathematicians, like M. Borel, rejected 
it resolutely, while others admitted it. Let us see what 
Mr. Russell thinks of it according to his last article. 

He pronounces no opinion, but the considerations 
which he gives are most suggestive. 

To begin with a picturesque example, suppose that 
we have as many pairs of boots as there are whole 
numbers, so that we can number the pairs from i to 
infinity, how many boots shall we have? Will the 
number of boots be equal to the number of pairs ? 
It will be so if, in each pair, the right boot is dis- 


tinguishable from the left ; it will be sufficient in fact to 
give the number 2« - i to the right boot of the n'" 
pair, and the number 2« to the left boot of the n'" 
pair. But it will not be so if the right boot is similar 
to the left, because such an operation then becomes 
impossible ; unless we admit Zermelo's axiom, since 
in that case we can select at random from each pair 
the boot we regard as the right. 



A demonstration really based upon the principles of 
Analytical Logic will be composed of a succession of 
propositions ; some, which will serve as premises, will 
be identities or definitions ; others will be deduced 
from the former step by step ; but although the con- 
nexion between each proposition and the succeeding 
proposition can be grasped immediately, it is not 
obvious at a glance how it has been possible to pass 
from the first to the last, which we may be tempted 
to look upon as a new truth. But if we replace 
successively the various expressions that are used by 
their definitions, and if we pursue this operation to the 
furthest possible limit, there will be nothing left at the 
end but identities, so that all will be reduced to one 
immense tautology. Logic therefore remains barren, 
unless it is fertilized by intuition. 

This is what I wrote formerly. The logisticians 
assert the contrary, and imagine that they have proved 
it by effectively demonstrating new truths. But what 
mechanism have they used ? 

(1,777) 13 


Why is it that by applying to their arguments the 
procedure I have just described, that is, by replacing 
the terms defined by their definitions, we do not see 
them melt into identities like the ordinary arguments ? 
It is because the procedure is not applicable to them. 
And why is this? Because their definitions are non- 
predicative and present that kind of hidden vicious 
circle I have pointed out above, and non-predicative 
definitions cannot be substituted for the term defined. 
Under these conditions. Logistic is no longer barren, it 
engenders antinomies. 

It is the belief in the existence of actual infinity that 
has given birth to these non-predicative definitions. I 
must explain myself. In these definitions we find the 
word all, as we saw in the examples quoted above. 
The word all has a very precise meaning when it is a 
question of a finite * number of objects ; but for it still 
to have a precise meaning when the number of the 
objects is infinite, it is necessary that there should 
exist an actual infinity. Otherwise all these objects 
cannot be conceived as existing prior to their definition, 
and then, if the definition of a notion N depends on 
all the objects A, it may be tainted with the vicious 
circle, if among the objects A there is one that cannot 
be defined without bringing in the notion N itself. 

The rules of formal logic simply express the pro- 
perties of all the possible classifications. But in order 
that they should be applicable, it is necessary that 
these classifications should be immutable and not 
require to be modified in the course of the argument. 
If we have only to classify a finite number of objects, 
it is easy to preserve these classifications without 

* The original has " infinite,'' obviously a slip. 


change. If the number of the objects is indefinite, 
that is to say if we are constantly liable to find new 
and unforeseen objects springing up, it may happen 
that the appearance of a new object will oblige us to 
modify the classification, and it is thus that we are 
exposed to the antinomies. 

There is no actual infinity. The Cantorians forgot 
this, and so fell into contradiction. It is true that 
Cantorism has been useful, but that was when it was 
applied to a real problem, whose terms were clearly 
defined, and then it was possible to advance without 

Like the Cantorians, the logisticians have forgotten 
the fact, and they have met with the same difficulties. 
But it is a question whether they took this path by 
accident or whether it was a necessity for them, 

In my view, there is no doubt about the matter ; 
belief in an actual infinity is essential in the Russellian 
logistic, and this is exactly what distinguishes it from 
the Hilbertian logistic. Hilbert takes the point of 
view of extension precisely in order to avoid the 
Cantorian antinomies. Russell takes the point of 
view of comprehension, and consequently for him the 
genus is prior to the species, and the summuni genus 
prior to all. This would involve no difficulty if the 
sunimum genus were finite ; but if it is infinite, it is 
necessary to place the infinite before the finite — that is 
to say, to regard the infinite as actual. 

And we have not only infinite classes ; when we 
pass from the genus to the species by restricting the 
concept by new conditions, the number of these 
conditions is still infinite, for they generally express 
that the object under consideration is in such and 


such a relation with all the objects of an infinite 

But all this is ancient history. Mr. Russell has 
realized the danger and is going to reconsider the 
matter. He is going to change everything, and we 
must understand clearly that he is preparing not only 
to introduce new principles which permit of operations 
formerly prohibited, but also to prohibit opera- 
tions which he formerly considered legitimate. He 
is not content with adoring what he once burnt, but 
he is going to burn what he once adored, which is 
more serious. He is not adding a new wing to the 
building, but sapping its foundations. 

The old Logistic is dead, and so true is this, that 
the zigzag theory and the no classes theory are 
already disputing the succession. We will wait until 
the new exists before we attempt to judge it. 






Are the general principles of Dynamics, which have 
served since Newton's day as the foundation of Physi- 
cal Science, and appear immutable, on the point of 
being abandoned, or, at the very least, profoundly 
modified ? This is the question many people have 
been asking for the last few years. According to 
them the discovery of radium has upset what were 
considered the most firmly rooted scientific doctrines, 
the impossibility of the transmutation of metals on the 
one hand, and, on the other, the fundamental postu- 
lates of Mechanics. Perhaps they have been in too 
great haste to consider these novelties as definitely 
established, and to shatter our idols of yesterday ; 
perhaps it would be well to await more numerous 
and more convincing experiments. It is none the less 
necessary that we should at once acquire a knowledge 
of the new doctrines and of the arguments, already 
most weighty, upon which they rely. 

I will first recall in a few words what these prin- 
ciples are. 


A. The motion of a material point, isolated and un- 
affected by any exterior force, is rectilineal and 
uniform. This is the principle of inertia ; no accelera- 
tion without force. 

B. The acceleration of a moving point has the same 
direction as the resultant of all the forces to which the 
point is subjected ; it is equal to the quotient of this 
resultant by a coefficient called the mass of the moving 

The mass of a moving point, thus defined, is con- 
stant; it does not depend upon the velocity acquired by 
the point, it is the same whether the force is parallel 
to this velocity and only tends to accelerate or retard 
the motion of the point, or whether it is, on the con- 
trary, perpendicular to that velocity and tends to 
cause the motion to deviate to right or left, that is to 
say to airve the trajectory. 

C. All the forces to which a material point is sub- 
jected arise from the action of other material points ; 
they depend only upon the relative positions and 
velocities of these different material points. 

By combining the two principles B and C we 
arrive at the principle of relative motion, by virtue of 
which the laws of motion of a system are the same 
whether we refer the system to fixed axes, or whether 
we refer it to moving axes animated with a rectilineal 
and uniform forward motion, so that it is impossible 
to distinguish absolute motion from a relative motion 
referred to such moving axes. 

D. If a material point A acts upon another material 
point B, the body B reacts upon A, and these two 
actions are two forces that are equal and directly 
opposite to one another. This is the principle of the 


equality of action and reaction, or more briefly, the 
principle of reaction. 

Astronomical observations, and the commonest 
physical phenomena, seem to have afforded the most 
complete, unvarying, and precise confirmation of these 
principles. That is true, they tell us now, but only 
because we have never dealt with any but low velo- 
cities. Mercury, for instance, which moves faster than 
any of the other planets, scarcely travels sixty miles a 
second — Would it behave in the same way if it travelled 
a thousand times as fast? It is clear that we have still 
no cause for anxiety ; whatever may be the progress 
of automobilism, it will be some time yet before we 
have to give up applying the classical principles of 
Dynamics to our machines. 

How is it then that we have succeeded in realizing 
velocities a thousand times greater than that of 
Mercury, equal, for instance, to a tenth or a third of 
the velocity of light, or coming nearer to it even than 
that? It is by the help of the cathode rays and 
the rays of radium. 

We know that radium emits three kinds of rays, 
which are designated by the three Greek letters «, ^, y. 
In what follows, unless I specifically state the contrary, 
I shall always speak of the fi rays, which are analogous 
to the cathode rays. 

After the discovery of the cathode rays, two opposite 
theories were propounded. Crookes attributed the 
phenomena to an actual molecular bombardment, 
Hertz to peculiar undulations of the ether. It was a 
repetition of the controversy that had divided physi- 
cists a century before with regard to light. Crookes 
returned to the emission theory, abandoned in the case 


of light, while Hertz held to the undulatory theory. 
The facts seemed to be in favour of Crookes. 

It was recognized in the first place that the cathode 
rays carry with them a negative electric charge : they 
are deviated by a magnetic and by an electric field, 
and these deviations are precisely what would be pro- 
duced by these same fields upon projectiles animated 
with a very great velocity, and highly charged with 
negative electricity. These two deviations depend 
upon two quantities ; the velocity on the one hand, 
and the proportion of the projectile's electric charge to 
its mass on the other. We cannot know the absolute 
value of this mass, nor that of the charge, but only 
their proportion. It is clear in fact, that if we double 
both the charge and the mass, without changing the 
velocity, we shall double the force that tends to deviate 
the projectile ; but as its mass is similarly doubled, 
the observable acceleration and deviation will not be 
changed. Observation of the two deviations will 
accordingly furnish us with two equations for deter- 
mining these two unknown quantities. We find a 
velocity of 6,000 to 20,000 miles a second. As for 
the proportion of the charge to the mass, it is very 
great ; it may be compared with the corresponding 
proportion in the case of a hydrogen ion in electro- 
lysis, and we find then that a cathode projectile 
carries with it about a thousand times as much 
electricity as an equal mass of hydrogen in an 

In order to confirm these views, we should require a 
direct measure of this velocity, that could then be 
compared with the velocity so calculated. Some old 
experiments of Sir J. J. Thomson's had given results 


more than a hundred times too low, but they were 
subject to certain causes of error. The question has 
been taken up again by Wiechert, with the help of an 
arrangement by which he makes use of the Hertzian 
oscillations, and this has given results in accordance 
with the theory, at least in the matter of magnitude, 
and it would be most interesting to take up these 
experiments again. However it be, the theory of 
undulations seems to be incapable of accounting for 
this body of facts. 

The same calculations made upon the /3 rays of 
radium have yielded still higher velocities — 60,000, 
120,000 miles a second, and even more. These 
velocities greatly surpass any that we know. It is 
true that light, as we have long known, travels 186,000 
miles a second, but it is not a transportation of matter, 
while, if we adopt the emission theory for the cathode 
rays, we have material molecules actually animated 
with the velocities in question, and we have to enquire 
whether the ordinary laws of Mechanics are still 
applicable to them. 


Longitudinal and Transversal Mass. 

We know that electric currents give rise to pheno- 
mena of induction, in particular to self-induction. 
When a current increases it develops an electro-motive 
force of self-induction which tends to oppose the 
current. On the contrary, when the current decreases, 
the electro-motive force of self-induction tends to 
maintain the current. Self-induction then opposes 
all variation in the intensity of a current, just as in 


Mechanics, the inertia of a body opposes all variation 
in its velocity. Self-induction is an actual inertia. 
Everything takes place as if the current could not be 
set up without setting the surrounding ether in motion, 
and as if the inertia of this ether consequently tended 
to keep the intensity of the current constant. The 
inertia must be overcome to set up the current, and it 
must be overcome again to make it cease. 

A cathode ray, which is a rain of projectiles charged 
with negative electricity, can be likened to a current. 
No doubt this current differs, at first sight at any rate, 
from the ordinary conduction currents, where the 
matter is motionless and the electricity circulates 
through the matter. It is a convection current, where 
the electricity is attached to a material vehicle and 
carried by the movement of that vehicle. But Rowland 
has proved that convection currents produce the same 
magnetic effects as conduction currents. They must 
also produce the same effects of induction. Firstly, if 
it were not so, the principle of the conservation of 
energy would be violated ; and secondly, Cremien and 
Pender have employed a method in which these effects 
of induction are directly demonstrated. 

If the velocity of a cathode corpuscle happens to 
vary, the intensity of the corresponding current will 
vary equally, and there will be developed effects of 
self-induction which tend to oppose this variation. 
These corpuscles must therefore possess a double 
inertia, first their actual inertia, and then an apparent 
inertia due to self-induction, which produces the same 
effects. They will therefore have a total apparent 
mass, composed of their real mass and of a fictitious 
mass of electro-magnetic origin. Calculation shows 


that this fictitious mass varies with the velocity (when 
this is comparable with the velocity of light), and that 
the force of the inertia of self-induction is not the 
same when the velocity of the projectile is increased 
or diminished, as when its direction is changed, and 
accordingly the same holds good of the apparent total 
force of inertia. 

The total apparent mass is therefore not the same 
when the actual force applied to the corpuscle is 
parallel with its velocity and tends to accelerate its 
movement, as when it is perpendicular to the velocity 
and tends to alter its direction. Accordingly we must 
distinguish between the total longitudinal mass and the 
total transversal mass, and, moreover, these two total 
masses depend upon the velocity. Such are the 
results of Abraham's theoretical work. 

In the measurements spoken of in the last section, 
what was it that was determined by measuring 
the two deviations ? The velocity on the one hand, 
and on the other the proportion of the charge to the 
total transversal mass. Under these conditions, how 
are we to determine what are the proportions, in this 
total mass, of the actual mass and of the fictitious 
electro-magnetic mass? If we had only the cathode 
rays properly so called, we could not dream of doing 
so, but fortunately we have the rays of radium, whose 
velocity, as we have seen, is considerably higher. 
These rays are not all identical, and do not behave 
in the same way under the action of an electric and a 
magnetic field. We find that the electric deviation 
is a function of the magnetic deviation, and by re- 
ceiving upon a sensitive plate rays of radium that 
have been subjected to the action of the two fields, 


we can photograph the curve which represents the 
relation between these two deviations. This is what 
Kaufmann has done, and he has deduced the rela- 
tion between the velocity and the proportion of the 
charge to the total apparent mass, a proportion that 
we call e. 

We might suppose that there exist several kinds 
of rays, each characterized by a particular velocity, 
by a particular charge, and by a particular mass ; 
but this hypothesis is most improbable. What reason 
indeed could there be why all the corpuscles of the 
same mass should always have the same velocity ? It 
is more natural to suppose that the charge and 
the actual mass are the same for all the projectiles, 
and that they differ only in velocity. If the propor- 
tion « is a function of the velocity, it is not because 
the actual mass varies with the velocity, but, as the 
fictitious electro-magnetic mass depends upon that 
velocity, the total apparent mass, which is alone 
observable, must depend upon it also, even though 
the actual mass does not depend upon it but is 

Abraham's calculations make us acquainted with 
the law in accordance with which the fictitious mass 
varies as a function of the velocity, and Kaufmann's 
experiment makes us acquainted with the law of 
variation of the total mass. A comparison of these 
two laws will therefore enable us to determine the 
proportion of the actual mass to the total mass. 

Such is the method employed by Kaufmann to 
determine this proportion. The result is most sur- 
prising : the actual mass is nil. 

We have thus been led to quite unexpected con- 


ceptions. What had been proved only in the case 
of the cathode corpuscles has been extended to all 
bodies. What we call mass would seem to be nothing 
but an appearance, and all inertia to be of electro- 
magnetic origin. But if this be true, mass is no 
longer constant; it increases with the velocity: while 
apparently constant for velocities up to as much as 
600 miles a second, it grows thenceforward and be- 
comes infinite for the velocity of light. Transversal 
mass is no longer equal to longitudinal mass, but only 
about equal if the velocity is not too great. Principle 
B of mechanics is no longer true. 



At the point we have reached, this conclusion may 
seem premature. Can we apply to the whole of 
matter what has only been established for these 
very light corpuscles which are only an emanation 
of matter and perhaps not true matter? But before 
broaching this question, we must say a word about 
another kind of rays — I mean the canal-rays, Gold- 
stein's Kanaktrahlen. Simultaneously with the cathode 
rays charged with negative electricity, the cathode 
emits canal-rays charged with positive electricity. In 
general these canal-rays, not being repelled by the cath- 
ode, remain confined in the immediate neighbourhood 
of that cathode, where they form the " buff stratum " 
that is not very easy to detect. But if the cathode is 
pierced with holes and blocks the tube almost com- 
pletely, the canal-rays will be generated behind the 


cathode, in the opposite direction from that of the 
cathode rays, and it will become possible to study 
them. It is thus that we have been enabled to 
demon.strate their positive charge and to show that 
the magnetic and electric deviations still exist, as 
in the case of the cathode rays, though they are much 

Radium likewise emits rays similar to the canal- 
rays, and relatively very absorbable, which are called 
a rays. 

As in the case of the cathode rays, we can measure 
the two deviations and deduce the velocity and the 
proportion e. The results are less constant than in 
the case of the cathode rays, but the velocity is lower, 
as is also the proportion «. The positive corpuscles 
are less highly charged than the negative corpuscles ; 
or if, as is more natural, we suppose that the charges 
are equal and of opposite sign, the positive corpuscles 
are much larger. These corpuscles, charged some 
positively and others negatively, have been given the 
name of electrons* 


LoRENTz's Theory. 

But the electrons do not only give evidence of 
their existence in these rays in which they appear 

* The name is now applied only to the negative corpuscles, which 
seem to possess no actual mass and only a fictitious electro-magnetic 
mass, and not to the canal-rays, which appear to consist of ordinary 
chemical atoms positively charged, owing to the fact that they have 
lost one or more of the electrons they possess in their ordinary neutral 


to us animated with enormous velocities. We shall 
see them in very different parts, and it is they that 
explain for us the principal phenomena of optics and 
of electricity. The brilliant synthesis about which I 
am going to say a few words is due to Lorentz. 

Matter is entirely formed of electrons bearing enor- 
mous charges, and if it appears to us neutral, it is 
because the electrons' charges of opposite sign balance. 
For instance, we can picture a kind of solar system 
consisting of one great positive electron, about which 
gravitate numerous small planets which are negative 
electrons, attracted by the electricity of opposite sign 
with which the central electron is charged. The 
negative charges of these planets balance the positive 
charge of the sun, so that the algebraic sum of all 
these charges is nil. 

All these electrons are immersed in ether. The 
ether is everywhere identical with itself, and perturba- 
tions are produced in it, following the same laws as 
light or the Hertzian oscillations in empty space. 
Beyond the electrons and the ether there is nothing. 
When a luminous wave penetrates a part of the ether 
where the electrons are numerous, these electrons are 
set in motion under the influence of the perturbation 
of the ether, and then react upon the ether. This 
accounts for refraction, dispersion, double refraction, 
and absorption. In the same way, if an electron was 
set in motion for any reason, it would disturb the 
ether about it and give birth to luminous waves, and 
this explains the emission of light by incandescent 

In certain bodies — metals, for instance — we have 
motionless electrons, about which circulate movable 

(1,777) 14 


electrons, enjoying complete liberty, except of leaving 
the metallic body and crossing the surface that sepa- 
rates it from exterior space, or from the air, or from 
any other non-metallic body. These movable elec- 
trons behave then inside the metallic body as do the 
molecules of a gas, according to the kinetic theory of 
gases, inside the vessel in which the gas is contained. 
But under the influence of a difference of potential 
the negative movable electrons would all tend to go 
to one side and the positive movable electrons to the 
other. This is what produces electric currents, and it 
is for this reason that such bodies act as conductors. 
Moreover, the velocities of our electrons will become 
greater as the temperature rises, if we accept the 
analogy of the kinetic theory of gases. When one 
of these movable electrons meets the surface of the 
metallic body, a surface it cannot cross, it is deflected 
like a billiard ball that has touched the cushion, and 
its velocity undergoes a sudden change of direction. 
But when an electron changes its direction, as we 
shall see further on, it becomes the source of a lumin- 
ous wave, and it is for this reason that hot metals are 

In other bodies, such as dielectric and transparent 
bodies, the movable electrons enjoy much less liberty. 
They remain, as it were, attached to fixed electrons 
which attract them. The further they stray, the 
greater becomes the attraction that tends to bring 
them back. Accordingly they can only suffer slight 
displacements ; they cannot circulate throughout the 
body, but only oscillate about their mean position. 
It is for this reason that these bodies are non- 
conductors ; they are, moreover, generally trans- 


parent, and they are refractive because the luminous 
vibrations are communicated to the movable electrons 
which are susceptible of oscillation, and a refraction 
of the original beam of light results. 

I cannot here give the details of the calculations. 
I will content myself with saying that this theory 
accounts for all the known facts, and has enabled us 
to foresee new ones, such as Zeeman's phenomenon. 



Now we can form two hypotheses in explanation of 
the above facts. 

1. The positive electrons possess an actual mass, 
much greater than their fictitious electro-magnetic 
mass, and the negative electrons alone are devoid of 
actual mass. We may even suppose that, besides the 
electrons of both signs, there are neutral atoms which 
have no other mass than their actual mass. In this 
case Mechanics is not affected, we have no need to 
touch its laws, actual mass is constant, only the move- 
ments are disturbed by the effects of self-induction, as 
has always been known. These perturbations are, 
moreover, almost negligible, except in the case of the 
negative electrons which, having no actual mass, are 
not true matter. 

2. But there is another point of view. We may sup- 
pose that the neutral atom does not exist, and that the 
positive electrons are devoid of actual mass just as 
much as the negative electrons. But if this be so, 
actual mass disappears, and either the word mass will 


have no further meaning, or else it must designate the 
fictitious electro-magnetic mass ; in that case mass will 
no longer be constant, transversal mass will no longer 
be equal to longitudinal mass, and the principles of 
Mechanics will be upset. 

And first a word by way of explanation. I said 
that, for the same charge, the total mass of a positive 
electron is much greater than that of a negative electron. 
Then it is natural to suppose that this difference is 
explained by the fact that the positive electron has, 
in addition to its fictitious mass, a considerable actual 
mass, which would bring us back to the first hypothesis. 
But we may equally well admit that the actual mass 
is nil for the one as for the other, but that the fictitious 
mass of the positive electron is much greater, because 
this electron is much smaller. I say advisedly, much 
smaller. And indeed, in this hypothesis, inertia is of 
exclusively electro-magnetic origin, and is reduced to 
the inertia of the ether ; the electrons are no longer 
anything in themselves, they are only holes in the 
ether, around which the ether is agitated ; the smaller 
these holes are, the more ether there will be, and the 
greater, consequently, will be its inertia. 

How are we to decide between these two hypotheses ? 
By working upon the canal-rays, as Kaufmann has 
done upon the /? rays? This is impossible, for the 
velocity of these rays is much too low. So each must 
decide according to his temperament, the conservatives 
taking one side and the lovers of novelty the other. 
But perhaps, to gain a complete understanding of 
the innovators' arguments, we must turn to other 




We know the nature of the phenomenon of aberration 
discovered by Bradley. The light emanating from a 
star takes a certain time to traverse the telescope. 
During this time the telescope is displaced by the 
Earth's motion. If, therefore, the telescope were 
pointed in the tme direction of the star, the image 
would be formed at the point occupied by the crossed 
threads of the reticule when the light reached the 
object-glass. When the light reached the plane of the 
reticule the crossed threads would no longer be in the 
same spot, owing to the Earth's motion. We are there- 
fore obliged to alter the direction of the telescope to 
bring the image back to the crossed threads. It 
follows that the astronomer will not point his telescope 
exactly in the direction of the absolute velocity of the 
light from the star — that is to say, upon the true position 
of the star — but in the direction of the relative velocity 
of the light in relation to the Earth — that is to say, upon 
what is called the apparent position of the star. 
The velocity of light is known, and accordingly we 


might imagine that we have the means of calculating 
the absolute velocity of the Earth. (I shall explain the 
meaning of this word "absolute" later.) But it is not 
so at all. We certainly know the apparent position of 
the star we are observing, but we do not know its true 
position. We know the velocity of light only in terms 
of magnitude and not of direction. 

If, therefore, the Earth's velocity were rectilineal and 
uniform, we should never have suspected the pheno- 
menon of aberration. But it is variable : it is composed 
of two parts — the velocity of the Solar System, which 
is, as far as we know, rectilineal and uniform ; and the 
velocity of the Earth in relation to the Sun, which is 
variable. If the velocity of the Solar System — that is 
to say the constant part — alone existed, the observed 
direction would be invariable. The position we should 
thus observe is called the mean apparent position of 
the star. 

Now if we take into account at once both parts of 
the Earth's velocity, we shall get the actual apparent 
position, which describes a small ellipse about the 
mean apparent position, and it is this ellipse that is 

Neglecting very small quantities, we shall see that 
the dimensions of this ellipse depend only upon the 
relation between the Earth's velocity in relation to the 
Sun and the velocity of light, so that the relative 
velocity of the Earth in relation to the Sun is alone 
in question. 

We must pause, however. This result is not exact, 
but only approximate. Let us push the approxima- 
tion a step further. The dimensions of the ellipse will 
then depend upon the absolute velocity of the Earth. 


If we compare the great axes of ellipse for the different 
stars, we shall have, theoretically at least, the means 
determining this absolute velocity. 

This is perhaps less startling than it seems at first. 
It is not a question, indeed, of the velocity in relation 
to absolute space, but of the velocity in relation to the 
ethics, which is regarded, by definition, as being in 
absolute repose. 

Moreover, this method is purely theoretical. In fact 
the aberration is very small, and the possible variations 
of the ellipse of aberration are much smaller still, and, 
acccordingly, if we regard the aberration as of the first 
order, the variations must be regarded as of the second 
order, about a thousandth of a second of arc, and 
absolutely inappreciable by our instruments. Lastly, 
we shall see further on why the foregoing theory must 
be rejected, and why we could not determine this 
absolute velocity even though our instruments were 
ten thousand times as accurate. 

Another method may be devised, and, indeed, has 
been devised. The velocity of light is not the same in 
the water as in the air : could we not compare the two 
apparent positions of a star seen through a telescope 
filled first with air and then with water ? The results 
have been negative ; the apparent laws of reflection 
and of refraction are not altered by the Earth's motion. 
This phenomenon admits of two explanations. 

I. We may suppose that the ether is not in repose, 
but that it is displaced by bodies in motion. It would 
not then be astonishing that the phenomenon of re- 
fraction should not be altered by the Earth's motion, 
since everything — lenses, telescopes, and ether — would 
be carried along together by the same motion. As for 


aberration itself, it would be explained by a kind of 
refraction produced at the surface of separation of the 
ether in repose in the interstellar spaces and the ether 
carried along by the Earth's movement. It is upon 
this hypothesis (the total translation of the ether) that 
Hertz's theory of the Electro-dynamics of bodies in 
motion is founded. 

2. Fresnel, on the contrary, supposes that the ether 
is in absolute repose in space, and almost in absolute 
repose in the air, whatever be the velocity of that air, 
and that it is partially displaced by refringent mediums. 
Lorentz has given this theory a more satisfactory form. 
In his view the ether is in repose and the electrons 
alone are in motion. In space, where the ether alone 
comes into play, and in the air, where it comes almost 
alone into play, the displacement is nil or almost nil. 
In refringent mediums, where the perturbation is pro- 
duced both by the vibrations of the ether and by 
those of the electrons set in motion by the agitation of 
the ether, the undulations 2X& partially carried along. 

To help us to decide between these two hypotheses, 
we have the experiment of Fizeau, who compared, by 
measurements of fringes of interference, the velocity of 
light in the air in repose and in motion as well as in 
water in repose and in motion. These experiments 
have confirmed Fresnel's hypothesis of partial dis- 
placement, and they have been repeated with the 
same result by Michelson. Hertz's theory, tlierefore, 
must be rejected. 


The Principle of Relativity. 

But if the ether is not displaced by the Earth's 
motion, is it possible by means of optical phenomena 
to demonstrate the absolute velocity of the Earth, or 
rather its velocity in relation to the motionless ether ? 
Experience has given a negative reply, and yet the 
experimental processes have been varied in every 
possible way. Whatever be the method employed, 
we shall never succeed in disclosing any but relative 
velocities ; I mean the velocities of certain material 
bodies in relation to other material bodies. Indeed, 
when the source of the light and the apparatus for 
observation are both on the Earth and participate in 
its motion, the experimental results have always been 
the same, whatever be the direction of the apparatus 
in relation to the direction of the Earth's orbital motion. 
That astronomical aberration takes place is due to the 
fact that the source, which is a star, is in motion in 
relation to the observer. 

The hypotheses formed up to now account perfectly 
for this general result, if we neglect very small quanti- 
ties on the order of the square of aberration. The 
explanation relies on the notion o{ local time introduced 
by Lorentz, which I will try to make clear. Imagine 
two observers placed, one at a point A and the other 
at a point B, wishing to set their watches by means of 
optical signals. They agree that B shall send a signal 
to A at a given hour by his watch, and A sets his 
watch to that hour as soon as he sees the signal. If 
the operation were performed in this way only, there 


would be a systematic error ; for, since light takes a 
certain time, t, to travel from B to A, A's watch would 
always be slower than B's to the extent of /. This 
error is easily corrected, for it is sufficient to inter- 
change the signals. A in his turn must send signals 
to B, and after this new setting it will be B's watch 
that will be slower than A's to the extent of t. Then 
it will only be necessary to take the arithmetic mean 
between the two settings. 

But this method of operating assumes that light 
takes the same time to travel from A to B and to 
return from B to A. This is true if the observers are 
motionless, but it is no longer true if they are involved 
in a common transposition, because in that case A, for 
instance, will be meeting the light that comes from B, 
while B is retreating from the light that comes from 
A. Accordingly, if the observers are involved in a 
common transposition without suspecting it, their set- 
ting will be defective ; their watches will not show the 
same time, but each of them will mark the local time 
proper to the place where it is. 

The two observers will have no means of detecting 
this, if the motionless ether can only transmit luminous 
signals all travelling at the same velocity, and if the 
other signals they can send are transmitted to them 
by mediums involved with them in their transposition. 
The phenomenon each of them observes will be either 
early or late — it will not occur at the moment it would 
have if there were no transposition ; but since their 
observations are made with a watch defectively set, 
they will not detect it, and the appearances will not 
be altered. 

It follows from this that the compensation is easy to 


explain so long as we neglect the square of aberration, 
and for a long time experiments were not sufficiently 
accurate to make it necessary to take this into account. 
But one day Michelson thought out a much more 
delicate process. He introduced rays that had 
traversed different distances after being reflected by 
mirrors. Each of the distances being about a yard, 
and the fringes of interference making it possible to 
detect differences of a fraction of a millionth of a 
millimeter (2-riinnnnTTrth of an inch), the square of 
aberration could no longer be neglected, and yet the 
results were still negative. Accordingly, the theory 
required to be completed, and this has been done by 
the hypothesis of Lorentz and fitz-Gerald. 

These two physicists assume that all bodies in- 
volved in a transposition undergo a contraction in the 
direction of this transposition, while their dimensions 
perpendicular to the transposition remain invariable. 
This cont7'action is the same for all bodies. It is, more- 
over, very slight, about one part in two hundred million 
for a velocity such as that of the Earth. Moreover, 
our measuring instruments could not disclose it, even 
though they were very much more accurate, since 
indeed the yard-measures with which we measure 
undergo the same contraction as the objects to be 
measured. If a body fits exactly to a measure when 
the body, and consequently the measure, are turned in 
the direction of the Earth's motion, it will not cease to 
fit exactly to the measure when turned in another 
direction, in spite of the fact that the body and the 
measure have changed their length in changing their 
direction, precisely because the change is the same for 
both. But it is not so if we measure a distance, no 


longer with a yard-measure, but by the time light 
takes to traverse it, and this is exactly what 
Michelson has done. 

A body that is spherical when in repose will thus 
assume the form of a flattened ellipsoid of revolution 
when it is in motion. But the observer will always 
believe it to be spherical, because he has himself under- 
gone an analogous deformation, as well as all the 
objects that serve him as points of reference. On the 
contrary, the surfaces of the waves of light, which have 
remained exactly spherical, will appear to him as 
elongated ellipsoids. 

What will happen then ? Imagine an observer and 
a source involved together in the transposition. The 
wave surfaces emanating from the source will be 
spheres, having as centre the successive positions of 
the source. The distance of this centre from the actual 
position of the source will be proportional to the time 
elapsed since the emission — that is to say, to the radius 
of the sphere. All these spheres are accordingly 
homothetic one to the other, in relation to the actual 
position S of the source. But for our observer, on 
account of the contraction, all these spheres will 
appear as elongated ellipsoids, and all these ellip- 
soids will still be homothetic in relation to the point 
S ; the excentricity of all the ellipsoids is the 
same, and depends solely upon the Earth's velocity. 
We shall select our law of contraction in such a way 
that S zvill be the focus of the meridian section of the 

This time the compensation is exact, and this is 
explained by Michelson's experiments. 

I said above that, according to the ordinary theories, 


observations of astronomical aberration could make us 
acquainted with the absolute velocity of the Earth, if 
our instruments were a thousand times as accurate, 
but this conclusion must be modified. It is true that 
the angles observed would be modified by the effect of 
this absolute velocity, but the graduated circles we use 
for measuring the angles would be deformed by the 
motion ; they would become ellipses, the result would 
be an error in the angle measured, and this second 
error would exactly compensate the forjner. 

This hypothesis of Lorentz and Fitz-Gerald will 
appear most extraordinary at first sight. All that can 
be said in its favour for the moment is that it is merely 
the immediate interpretation of Michelson's experi- 
mental result, if we define distances by the time taken 
by light to traverse them. 

However that be, it is impossible to escape the 
impression that the Principle of Relativity is a general 
law of Nature, and that we shall never succeed, by any 
imaginable method, in demonstrating any but relative 
velocities ; and by this I mean not merely the velocities 
of bodies in relation to the ether, but the velocities ot 
bodies in relation to each other. So many different 
experiments have given similar results that we cannot 
but feel tempted to attribute to this Principle of 
Relativity a value comparable, for instance, to that of 
the Principle of Equivalence. It is well in any case to 
see what are the consequences to which this point of 
view would lead, and then to submit these consequences 
to the test of experiment. 


The Principle of Reaction. 

Let us see what becomes, under Lorentz's theory, 
of the principle of the equality of action and reaction. 
Take an electron. A, which ie set in motion by some 
means. It produces a disturbance in the ether, and 
after a certain time this disturbance reaches another 
electron, B, which will be thrown out of its posi- 
tion of equilibrium. Under these conditions there 
can be no equality between the action and the re- 
action, at least if we do not consider the ether, but 
only the electrons which are alone observable, since 
our matter is composed of electrons. 

It is indeed the electron A that has disturbed the 
electron B ; but even if the electron B reacts upon A, 
this reaction, though possibly equal to the action, 
cannot in any case be simultaneous, since the electron 
B cannot be set in motion until after a certain length 
of time necessary for the effect to travel through the 
ether. If we submit the problem to a more precise 
calculation, we arrive at the following result. Imagine 
a Hertz excitator placed at the focus of a parabolic 
mirror to which it is attached mechanically ; this 
excitator emits electro-magnetic waves, and the mirror 
drives all these waves in the same direction : the 
excitator will accordingly radiate energy in a particular 
direction. Well, calculations show that the excitator 
will recoil like a cannon that has fired a projectile. 
In the case of the cannon, the recoil is the natural 
result of the equality of action and reaction. The 


cannon recoils because the projectile on which it has 
acted reacts upon it. 

But here the case is not the same. What we have 
fired away is no longer a material projectile ; it is 
energy, and energy has no mass — there is no counter- 
part. Instead of an excitator, we might have con- 
sidered simply a lamp with a reflector concentrating 
its rays in a single direction. 

It is true that if the energy emanating from the 
excitator or the lamp happens to reach a material 
object, this object will experience a mechanical thrust 
as if it had been struck by an actual projectile, and 
this thrust will be equal to the recoil of the excitator 
or the lamp, if no energy has been lost on the way, 
and if the object absorbs the energy in its entirety. 
We should then be tempted to say that there is still 
compensation between the action and the reaction. 
But this compensation, even though it is complete, is 
always late. It never occurs at all if the light, after 
leaving the source, strays in the interstellar spaces 
without ever meeting a material body, and it is 
incomplete if the body it strikes is not perfectly 

Are these mechanical actions too small to be 
measured, or are they appreciable by experiment ? 
They are none other than the actions due to the 
Maxwell-Bartholi pressures. Maxwell had predicted 
these pressures by calculations relating to Electro- 
statics and Magnetism, and Bartholi had arrived at 
the same results on Thermodynamic grounds. 

It is in this way that tails of comets are explained. 
Small particles are detached from the head of the 
comet, they are struck by the light of the Sun, which 


repels them just as would a shower of projectiles 
coming from the Sun. The mass of these particles is 
so small that this repulsion overcomes the Newtonian 
gravitation, and accordingly they form the tail as they 
retreat from the Sun. 

Direct experimental verification of this pressure ot 
radiation was not easy to obtain. The first attempt 
led to the construction of the radiometer. But this 
apparatus turns the wrong way, the reverse of the 
theoretical direction, and the explanation of its rota- 
tion, which has since been discovered, is entirely 
different. Success has been attained at last by creat- 
ing a more perfect vacuum on the one hand ; and 
on the other, by not blackening one of the faces of 
the plates, and by directing a luminous beam upon 
one of these faces. The radiometric effects and other 
disturbing causes are eliminated by a series of minute 
precautions, and a deviation is obtained which is 
extremely small, but is, it appears, in conformity with 
the theory. 

The same effects of the Maxwell-Bartholi pressure 
are similarly predicted by Hertz's theory, of which I 
spoke above, and by that of Lorentz, but there is a 
difference. Suppose the energy, in the form of light, 
for instance, travels from a luminous source to any 
body through a transparent medium. The Maxwell- 
Bartholi pressure will act not only upon the source at 
its start and upon the body lighted at its arrival, but 
also upon the matter of the transparent medium it 
traverses. At the moment the luminous wave reaches 
a new portion of this medium, the pressure will drive 
forward the matter there distributed, and will drive it 
back again when the wave leaves that portion. So 


that the recoil of the source has for its counterpart the 
forward motion of the transparent matter that is in 
contact with the source ; a Httle later the recoil of 
this same matter has for its counterpart the forward 
motion of the transparent matter a little further off, 
and so on. 

Only, is the compensation perfect ? Is the action of 
the Maxwell-Bartholi pressure upon the matter of the 
transparent medium equal to its reaction upon the 
source, and that, whatever that matter may be ? Or 
rather, is the action less in proportion as the medium 
is less refringent and more rarefied, becoming nil in a 
vacuum ? If we admit Hertz's theory, which regards 
the ether as mechanically attached to matter, so that 
the ether is completely carried along by matter, we 
must answer the first and not the second question in 
the affirmative. 

There would then be perfect compensation, such as 
the principle of the equality of action and reaction 
demands, even in the least refringent media, even in 
the air, even in the interplanetary space, where it 
would be sufficient to imagine a bare remnant of 
matter, however attenuated. If we admit Lorentz's 
theory, on the contrary, the compensation, always 
imperfect, is inappreciable in the air, and becomes nil 
in space. 

But we have seen above that Fizeau's experiment 
does not permit of our retaining Hertz's theory. We 
must accordingly adopt Lorentz's theory, and conse- 
quently give up the principle of reaction. 

(1,777) 15 



Consequences of the Principle of 

We have seen above the reasons that incline us to 
regard the Principle of Relativity as a general law of 
Nature. Let us see what consequences the principle 
will lead us to if we regard it as definitely proved. 

First of all, it compels us to generalize the hypo- 
thesis of Lorentz and Fitz-Gerald on the contraction 
of all bodies in the direction of their transposition. 
More particularly, we must extend the hypothesis to 
the electrons themselves. Abraham considered these 
electrons as spherical and undeformable, but we shall 
have to admit that the electrons, while spherical when 
in repose, undergo Lorentz's contraction when they 
are in motion, and then take the form of flattened 

This deformation of the electrons will have an 
influence upon their mechanical properties. In fact, 
I have said that the displacement of these charged 
electrons is an actual convection current, and that 
their apparent inertia is due to the self-induction of 
this current, exclusively so in the case of the negative 
electrons, but whether exclusively or not in the case of 
the positive electrons we do not yet know. 

On these terms the compensation will be perfect, 
and in conformity with the requirements of the 
Principle of Relativity, but only upon two con- 
ditions : — 

I. That the positive electrons have no real mass, 
but only a fictitious electro-magnetic mass ; or at least 


that their real mass, if it exists, is not constant, but 
varies with the velocity, following the same laws as 
their fictitious mass. 

2. That all forces are of electro-magnetic origin, or 
at least that they vary with the velocity, following the 
same laws as forces of electro-magnetic origin. 

It is Lorentz again who has made this remarkable 
synthesis. Let us pause a moment to consider what 
results from it. In the first place, there is no more 
matter, since the positive electrons have no longer 
any real mass, or at least no constant real mass. The 
actual principles of our Mechanics, based upon the 
constancy of mass, must accordingly be modified. 

Secondly, we must seek an electro-magnetic ex- 
planation of all known forces, and especially of gravi- 
tation, or at least modify the law of gravitation in the 
sense that this force must be altered by velocity in 
the same way as electro-magnetic forces. We shall 
return to this point. 

All this appears somewhat artificial at first sight, 
and more particularly the deformation of the electrons 
seems extremely hypothetical. But the matter can 
be presented differently, so as to avoid taking this 
hypothesis of deformation as the basis of the argu- 
ment. Let us imagine the electrons as material points, 
and enquire how their mass ought to vary as a function 
of the velocity so as not to violate the Principle of 
Relativity. Or rather let us further enquire what should 
be their acceleration under the influence of an electric 
or magnetic field, so that the principle should not be 
violated and that we should return to the ordinary 
laws when we imagine the velocity very low. We 
shall find that the variations of this mass or of these 


accelerations must occur as if the electron underwent 
Lorentz's deformation. 


Kaufmann's Experiment. 

Two theories are thus presented to us : one in 
which the electrons are undeformable, which is Abra- 
ham's ; the other, in which they undergo Lorentz's 
deformation. In either case their mass grows with 
their velocity, becoming infinite when that velocity 
becomes equal to that of light ; but the law of the 
variation is not the same. The method employed by 
Kaufmann to demonstrate the law of variation of the 
mass would accordingly seem to give us the means of 
deciding experimentally between the two theories. 

Unfortunately his first experiments were not suffi- 
ciently accurate for this purpose, so much so that he 
has thought it necessary to repeat them with more 
precautions, and measuring the intensity of the fields 
with greater care. In their new form they have shown 
Abraham's theory to be right. Accordingly, it would seem 
that the Principle of Relativity has not the exact value 
we have been tempted to give it, and that we have no 
longer any reason for supposing that the positive elec- 
trons are devoid of real mass like the negative electrons. 

Nevertheless, before adopting this conclusion some 
reflexion is necessary. The question is one of such 
importance that one would wish to see Kaufmann's 
experiment repeated by another experimenter.* 

* At the moment of going to press we learn that M. Bucherer has 
repeated the experiment, surrounding it with new precautions, and that, 
unlike Kaufmann, he has obtained results confirming Lorentz's views. 


Unfortunately, the experiment is a very delicate 
one, and cannot be performed successfully, except by 
a physicist as skilful as Kaufmann. All suitable pre- 
cautions have been taken, and one cannot well see 
what objection can be brought. 

There is, nevertheless, one point to which I should 
wish to call attention, and that is the measurement of 
the electrostatic field, the measurement upon which 
everything depends. This field was produced between 
the two armatures of a condenser, and between these 
two armatures an extremely perfect vacuum had to 
be created in order to obtain complete isolation. The 
difference in the potential of the two armatures was 
then measured, and the field was obtained by dividing 
this difference by the distance between the armatures. 
This assumes that the field is uniform ; but is this 
certain ? May it not be that there is a sudden drop 
in the potential in the neighbourhood of one of the 
armatures, of the negative armature, for instance? 
There may be a difference in potential at the point 
of contact between the metal and the vacuum, and it 
may be that this difference is not the same on the 
positive as on the negative side. What leads me to 
think this is the electric valve effect between mercury 
and vacuum. It would seem that we must at least 
take into account the possibility of this occurring, 
however slight the probability may be. 


The Principle of Inertia. 

In the new Dynamics the Principle of Inertia is still 
true — that is to say, that an isolated electron will have 


a rectilineal and uniform motion. At least it is gener- 
ally agreed to admit it, though Lindemann has raised 
objections to the assumption. I do not wish to take 
sides in the discussion, which I cannot set out here 
on account of its extremely difficult nature. In any 
case, the theory would only require slight modifications 
to escape Lindemann's objections. 

We know that a body immersed in a fluid meets 
with considerable resistance when it is in motion ; but 
that is because our fluids are viscous. In an ideal 
fluid, absolutely devoid of viscidity, the body would 
excite behind it a liquid stern-wave, a kind of wake. 
At the start, it would require a great effort to set it 
in motion, since it would be necessary to disturb not 
only the body itself but the liquid of its wake. But 
once the motion was acquired, it would continue 
without resistance, since the body, as it advanced, 
would simply carry with it the disturbance of the 
liquid, without any increase in the total vis viva of 
the liquid. Everything would take place, therefore, 
as if its inertia had been increased. An electron 
advancing through the ether will behave in the same 
way. About it the ether will be disturbed, but this 
disturbance will accompany the body in its motion, so 
that, to an observer moving with the electron, the 
electric and magnetic fields which accompany the 
electron would appear invariable, and could only 
change if the velocity of the electron happened to 
vary. An effort is therefore required to set the 
electron in motion, since it is necessary to create the 
energy of these fields. On the other hand, once the 
motion is acquired, no effort is necessary to maintain 
it, since the energy created has only to follow the 


electron like a wake. This energy, therefore, can only 
increase the inertia of the electron, as the agitation 
of the liquid increases that of the body immersed in 
a perfect fluid. And actually the electrons, at any 
rate the negative electrons, have no other inertia but 

In Lorentz's hypothesis, the vis viva, which is 
nothing but the energy of the ether, is not propor- 
tional to v^. No doubt if v is very small, the vis 
viva is apparently proportional to v"^, the amount of 
momentum apparently proportional to v, and the two 
masses apparently constant and equal to one another. 
But when the velocity approaches the velocity of light, 
the vis viva, the amount of momentum, and the two 
masses increase beyond all limit. 

In Abraham's hypothesis the expressions are some- 
what more complicated, but what has just been said 
holds good in its essential features. 

Thus the mass, the amount of momentum, and the 
vis viva become infinite when the velocity is equal to 
that of light. Hence it follows that no body can, by 
any possibility, attain a velocity higher than that of 
light. And, indeed, as its velocity increases its mass 
increases, so that its inertia opposes a more and more 
serious obstacle to any fresh increase in its velocity. 

A question then presents itself. Admitting the 
Principle of Relativity, an observer in motion can have 
no means of perceiving his own motion. If, therefore, 
no body in its actual motion can exceed the velocity 
of light, but can come as near it as we like, it must be the 
same with regard to its relative motion in relation to 
our observer. Then we might be tempted to reason 
as follows : — The observer can attain a velocity of 


120,000 miles a second, the body in its relative motion 
in relation to the observer can attain the same velocity ; 
its absolute velocity will then be 240,000 miles, which 
is impossible, since this is a figure higher than that of 
the velocity of light. But this is only an appearance 
which vanishes when we take into account Lorentz's 
method of valuing local times. 

The Wave of Acceleratlon. 

When an electron is in motion it produces a dis- 
turbance in the ether which surrounds it. If its 
motion is rectilineal and uniform, this disturbance is 
reduced to the wake I spoke of in the last section. 
But it is not so if the motion is in a curve or not 
uniform. The disturbance may then be regarded as 
the superposition of two others, to which Langevin 
has given the names of wave of velocity and wave of 

The wave of velocity is nothing else than the wake 
produced by the uniform motion. 

As for the wave of acceleration, it is a disturbance 
absolutely similar to light waves, which starts from 
the electron the moment it undergoes an acceleration, 
and is then transmitted in successive spherical waves 
with the velocity of light. 

Hence it follows that in a rectilineal and uniform 
motion there is complete conservation of energy, but 
as soon as there is acceleration there is loss of energy, 
which is dissipated in the form of light waves and 
disappears into infinite space through the ether. 


Nevertheless, the effects of this wave of acceleration, 
and more particularly the corresponding loss of energy, 
are negligible in the majority of cases — that is to say, 
not only in the ordinary Mechanics and in the motions 
of the celestial bodies, but even in the case of the radium 
rays, where the velocity, but not the acceleration, is 
very great. We may then content ourselves with the 
application of the laws of Mechanics, stating that the 
force is equal to the product of the acceleration and 
the mass, this mass, however, varying with the velocity 
according to the laws set forth above. The motion is 
then said to be quasi-stationary. 

It is not so in all the cases where the acceleration is 
great, the chief of which are as follows, (i.) In incan- 
descent gases certain electrons take on an oscillatory 
motion of very high frequency ; the displacements are 
very small, the velocities finite, and the accelerations 
very great ; the energy is then communicated to the 
ether, and it is for this reason that these gases radiate 
light of the same periodicity as the oscillations of the 
electron. (2.) Inversely, when a gas receives light, 
these same electrons are set in motion with violent 
accelerations, and they absorb light. (3.) In Hertz's 
excitator, the electrons which circulate in the metallic 
mass undergo a sudden acceleration at the moment of 
the discharge, and then take on an oscillatory motion 
of high frequency. It follows that a part of the energy 
is radiated in the form of Hertzian waves. (4.) In an 
incandescent metal, the electrons enclosed in the metal 
are animated with great velocities. On arriving at the 
surface of the metal, which they cannot cross, they are 
deflected, and so undergo a considerable acceleration, 
and it is for this reason that the metal emits light. 


This I have already explained in Book III., Chap. I., 
Sec. 4. The details of the laws of the emission of 
light by dark bodies are perfectly explained by this 
hypothesis. (5.) Lastly, when the cathode rays strike 
the anticathode, the negative electrons constituting 
these rays, which are animated with very great velo- 
cities, are suddenly stopped. In consequence of the 
acceleration they thus undergo, they produce undula- 
tions in the ether. This, according to certain 
physicists, is the origin of the Rontgen rays, which are 
nothing else than light rays of very short wave length. 





Mass may be defined in two ways — firstly, as the 
quotient of the force by the acceleration, the true 
definition of mass, which is the measure of the body's 
inertia ; and secondly, as the attraction exercised by 
the body upon a foreign body, by virtue of Newton's 
law. We have therefore to distinguish between mass, 
the coefficient of inertia, and mass, the coefficient of 
attraction. According to Newton's law, there is a 
rigorous proportion between these two coefficients, but 
this is only demonstrated in the case of velocities to 
which the general principles of Dynamics are appli- 
cable. Now we have seen that the mass coefficient of 
inertia increases with the velocity ; must we conclude 
that the mass coefficient of attraction increases 
similarly with the velocity, and remains proportional 
to the coefficient of inertia, or rather that the 
coefficient of attraction remains constant ? This is a 
question that we have no means of deciding. 

On the other hand, if the coefficient of attraction 
depends upon the velocity, as the velocities of bodies 


mutually attracting each other are generally not the 
same, how can this coefficient depend upon these two 
velocities ? 

Upon this subject wc can but form hypotheses, but 
we are naturally led to enquire which of these hypo- 
theses will be compatible with the Principle of 
Relativity. There are a great number, but the only 
one I will mention here is Lorentz's hypothesis, which 
I will state briefly. 

Imagine first of all electrons in repose. Two 
electrons of similar sign repel one another, and two 
electrons of opposite sign attract one another. Accord- 
ing to the ordinary theory, their mutual actions are 
proportional to their electric charges. If, therefore, 
we have four electrons, two positive, A and A', and 
two negative, B and B', and the charges of these four 
electrons are the same in absolute value, the repulsion 
of A upon A' will be, at the same distance, equal to 
the repulsion of B upon B', and also equal to the 
attraction of A upon B' or of A' upon B. Then if A and 
B are very close to each other, as also A' and B', and 
we examine the action of the system A -t- B upon 
the system A'-i-B', we shall have two repulsions and 
two attractions that are exactly compensated, and the 
resultant action will be nil. 

Now material molecules must precisely be regarded 
as kinds of solar systems in which the electrons circulate, 
some positive and others negative, in such a way that 
the algebraic sum of all the charges is nil. A material 
molecule is thus in all points comparable to the system 
A + B I have just spoken of, so that the total 
electric action of two molecules upon each other 
should be nil. 


But experience shows us that these molecules attract 
one another in accordance with Newtonian gravitation, 
and that being so we can form two hypotheses. We 
may suppose that gravitation has no connexion with 
electrostatic attraction, that it is due to an entirely 
different cause, and that it is merely superimposed 
upon it ; or else we may admit that there is no pro- 
portion between the attractions and the charges, and 
that the attraction exercised by a charge + 1 upon a 
charge - I is greater than the mutual repulsion of two 
charges + i or of two charges - i. 

In other words, the electric field produced by the 
positive electrons and that produced by the negative 
electrons are superimposed and remain distinct. The 
positive electrons are more sensitive to the field pro- 
duced by the negative electrons than to the field pro- 
duced by the positive electrons, and contrariwise for 
the negative electrons. It is clear that this hypothesis 
somewhat complicates electrostatics, but makes it 
include gravitation. It was, in the main, Franklin's 

jNow, what happens if the electrons are in motion ? 
The positive electrons will create a disturbance in the 
ether, and will give rise in it to an electric field and a 
magnetic field. The same will be true of the negative 
electrons. The electrons, whether positive or negative, 
then receive a mechanical impulse by the action of 
these different fields. In the ordinary theory, the 
electro-magnetic field due to the motion of the positive 
electrons exercises, upon two electrons of opposite 
sign and of the same absolute charge, actions that are 
equal and of opposite sign. We may, then, without 
impropriety make no distinction between the field due 


to the motion of the positive electrons and the field 
due to the motion of the negative electrons, and 
consider only the algebraic sum of these two fields^ 
that is to say, the resultant field. 

In the new theory, on the contrary, the action upon 
the positive electrons of the electro-magnetic field due 
to the positive electrons takes place in accordance 
with the ordinary laws, and the same is true of the 
action upon the negative electrons of the field due 
to the negative electrons. Let us now consider the 
action of the field due to the positive electrons upon 
the negative electrons, or vice versa. It will still 
follow the same laws, but with a different coefficient. 
Each electron is more sensitive to the field created 
by the electrons of opposite denomination than to 
the field created by the electrons of the same de- 

Such is Lorentz's hypothesis, which is reduced to 
Franklin's hypothesis for low velocities. It agrees 
with Newton's law in the case of these low velocities. 
More than that, as gravitation is brought down to 
forces of electro-dynamic origin, Lorentz's general 
theory will be applicable to it, and consequently the 
Principle of Relativity will not be violated. 

We see that Newton's law is no longer applicable to 
great velocities, and that it must be modified, for 
bodies in motion, precisely in the same way as the 
laws of Electrostatics have to be for electricity in 

We know that electro-magnetic disturbances are 
transmitted with the velocity of light. We shall 
therefore be tempted to reject the foregoing theory, 
remembering that gravitation is transmitted, according 


to Laplace's calculations, at least ten million times as 
quickly as light, and that consequently it cannot be of 
electro-magnetic origin. Laplace's result is well known, 
but its significance is generally lost sight of. Laplace 
assumed that, if the transmission of gravitation is not 
in.stantaneous, its velocity of transmission combines 
with that of the attracted body, as happens in the case 
of light in the phenomenon of astronomical aberration, 
in such a way that the effective force is not directed 
along the straight line joining the two bodies, but 
makes a small angle with that straight line. This is 
quite an individual hypothesis, not very well sub- 
stantiated, and in any case entirely different from that 
of Lorentz. Laplace's result proves nothing against 
Lorentz's theory. 


Comparison with Astronomical 

Are the foregoing theories reconcilable with astro- 
nomical observations ? To begin with, if we adopt 
them, the energy of the planetary motions will be 
constantly dissipated by the effect of the wave of 
acceleration. It would follow from this that there would 
be a constant acceleration of the mean motions of the 
planets, as if these planets were moving in a resisting 
medium. But this effect is exceedingly slight, much 
too slight to be disclosed by the most minute obser- 
vation.s. The acceleration of the celestial bodies is 
relatively small, so that the effects of the wave of 
acceleration are negligible, and the motion may be 
regarded as quasi-stationary. It is true that the 


effects of the wave of acceleration are constantly 
accumulating, but this accumulation itself is so slow 
that it would certainly require thousands of years of 
observation before it became perceptible. 

Let us therefore make the calculation, taking the 
motion as quasi-stationary, and that under the three 
following hypotheses : — 

A. Admitting Abraham's hypothesis (undeformable 
electrons), and retaining Newton's law in its ordinary 

B. Admitting Lorentz's hypothesis concerning the 
deformation of the electrons, and retaining Newton's 
ordinary law. 

C. Admitting Lorentz's hypothesis concerning the 
electrons, and modifying Newton's law, as in the fore- 
going section, so as to make it compatible with the 
Principle of Relativity. 

It is in the motion of Mercury that the effect will 
be most perceptible, because it is the planet that has 
the highest velocity. Tisserand formerly made a 
similar calculation, admitting Weber's law. I would 
remind the reader that Weber attempted to explain 
both the electrostatic and the electro-dynamic phe- 
nomena, assuming that the electrons (whose name had 
not yet been invented) exercise upon each other attrac- 
tions and repulsions in the direction of the straight 
line joining them, and depending not only upon their 
distances, but also upon the first and second deriva- 
tives of these distances, that is consequently upon 
their velocities and their accelerations. This law of 
Weber's, different as it is from those that tend to gain 
acceptance to-day, presents none the less a certain 
analogy with them. 


Tisserand found that if the Newtonian attraction 
took place in conformity with Weber's law, there would 
result, in the perihelion of Mercury, a secular variation 
of 14", in the same direction as that which has been 
observed and not explained, but smaller, since the 
latter is 38". 

Let us return to the hypotheses A, B, and C, and study 
first the motion of a planet attracted by a fixed centre. 
In this case there will be no distinction between 
hypotheses B and C, since, if the attracting point is 
fixed, the field it produces is a purely electrostatic 
field, in which the attraction varies in the inverse 
ratio of the square of the distance, in conformity with 
Coulomb's electrostatic law, which is identical with 

The vis viva equation holds good if we accept the 
new definition of vis viva. In the same way the 
equation of the areas is replaced by another equivalent. 
The moment of the quantity of motion is a constant, 
but the quantity of motion must be defined in the 
new way. 

The only observable effect will be a secular motion 
of the perihelion. For this motion we shall get, with 
Lorentz's theory, a half, and with Abraham's theory 
two-fifths, of what was given by Weber's law. 

If we now imagine two moving bodies gravitating 
about their common centre of gravity, the effects are 
but very slightly different, although the calculations 
are somewhat more complicated. The motion of 
Mercury's perihelion will then be 7" in Lorentz's 
theory, and 5.6" in Abraham's. 

The effect is, moreover, proportional to «V, n being 
the mean motion of the planet, and a the radius of its 
(1.777) 16 


orbit. Accordingly for the planets, by virtue of 
Kepler's law, the effect varies in the inverse ratio of 
sja^, and it is therefore imperceptible except in the 
case of Mercury. 

It is equally imperceptible in the case of the Moon, 
because, though n is large, a is extremely small. 
In short, it is five times as small for Venus, and six 
hundred times as small for the Moon, as it is for 
Mercury. I would add that as regards Venus and 
the Earth, the motion of the perihelion (for the same 
angular velocity of this motion) would be much more 
difficult to detect by astronomical observations, because 
the excentricity of their orbits is much slighter than in 
the case of Mercury. 

To sum up, the only appreciable effect upon astronom- 
ical observations would be a motion of Mercury's peri- 
helion, in the same direction as that which has been 
observed without being explained, but considerably 

This cannot be regarded as an argument in favour 
of the new Dynamics, since we still have to seek 
another explanation of the greater part of the anomaly 
connected with Mercury ; but still less can it be 
regarded as an argument against it. 


Lesage's Theory, 

It would be well to set these considerations beside 
a theory put forward long ago to explain universal 
gravitation. Imagine the interplanetary spaces full of 
very tiny corpuscles, travelling in all directions at very 


high velocities. An isolated body in space will not 
be affected apparently by the collisions with these 
corpuscles, since the collisions are distributed equally 
in all directions. But if two bodies, A and B, are in 
proximity, the body B will act as a screen, and inter- 
cept a portion of the corpuscles, which, but for it, 
would have struck A. Then the collisions received 
by A from the side away from B will have no counter- 
part, or will be only imperfectly compensated, and will 
drive A towards B. 

Such is Lesage's theory, and we will discuss it first 
from the point of view of ordinary mechanics. To begin 
with, how must the collisions required by this theory 
occur? Must it be in accordance with the laws of 
perfectly elastic bodies, or of bodies devoid of elasticity, 
or in accordance with some intermediate law ? Lesage's 
corpuscles cannot behave like perfectly elastic bodies, 
for in that case the effect would be nil, because the 
corpuscles intercepted by the body B would be replaced 
by others which would have rebounded from B, and 
calculation proves that the compensation would be 

The collision must therefore cause a loss of energy 
to the corpuscles, and this energy should reappear in 
the form of heat. But what would be the amount of 
heat so produced? We notice that the attraction 
passes through the body, and we must accordingly 
picture the Earth, for instance, not as a complete 
screen, but as composed of a very large number of 
extremely small spherical molecules, acting individually 
as little screens, but allowing Lesage's corpuscles to 
travel freely between them. Thus, not only is the 
Earth not a complete screen, but it is not even a 


strainer, since the unoccupied spaces are much larger 
than the occupied. To reahze this, we must remem- 
ber that Laplace demonstrated that the attraction, in 
passing through the Earth, suffers a loss, at the very- 
most, of a ten-millionth part, and his demonstration is 
perfectly satisfactory. Indeed, if the attraction were 
absorbed by the bodies it passes through, it would no 
longer be proportional to their masses ; it would be 
relatively weaker for large than for small bodies, since 
it would have a greater thickness to traverse. The 
attraction of the Sun for the Earth would therefore be 
relatively weaker than that of the Sun for the Moon, 
and a very appreciable inequality in the Moon's motion 
would result. We must therefore conclude, if we adopt 
Lesage's theory, that the total surface of the spherical 
molecules of which the Earth is composed is, at the 
most, the ten-millionth part of the total surface of the 

Darwin proved that Lesage's theory can only lead 
exactly to Newton's law if we assume the corpuscles 
to be totally devoid of elasticity. The attraction 
exercised by the Earth upon a mass i at a distance i 
will then be proportional both to S, the total surface 
of the spherical molecules of which it is composed, to 
f, the velocity of the corpuscles, and to the square 
root of p, the density of the medium formed by the 
corpuscles. The heat produced will be proportional 
to S, to the density p, and to the cube of the 
velocity v. 

But we must take account of the resistance ex- 
perienced by a body moving in such a medium. It 
cannot move, in fact, without advancing towards certain 
collisions, and on the other hand retreating before 


those that come from the opposite direction, so that 
the compensation realized in a state of repose no longer 
exists. The calculated resistance is proportional to S, 
to p, and to v. Now we know that the heavenly bodies 
move as if they met with no resistance, and the pre- 
cision of the observations enables us to assign a limit 
to the resistance. 

This resistance varying as Spv, while the attraction 
varies as S Jpv, we see that the relation of the resist- 
ance to the square of the attraction is in inverse ratio 
of the product Sv. 

We get thus an inferior limit for the product Sv. 
We had already a superior limit for S (by the absorp- 
tion of the attraction by the bodies it traverses). We 
thus get an inferior limit for the velocity v, which must 
be at least equal to 24.10" times the velocity of light. 

From this we can deduce p and the amount of heat 
produced. This would suffice to elevate the tempera- 
ture 10^ degrees a second. In any given time the 
Earth would receive io^° as much heat as the Sun 
emits in the same time, and I am not speaking of 
the heat that reaches the Earth from the Sun, but of 
the heat radiated in all directions. It is clear that 
the Earth could not long resist such conditions. 

We shall be led to results no less fantastic if, in 
opposition to Darwin's views, we endow Lesage's 
corpuscles with an elasticity that is imperfect but 
not nil. It is true that the vis viva of the corpuscles 
will not then be entirely converted into heat, but the 
attraction produced will equally be less, so that it 
will only be that portion of the vis viva converted 
into heat that will contribute towards the production 
of attraction, and so we shall get the same result. A 


judicious use of the theorem of virial will enable us 
to realize this. 

We may transform Lesage's theory by suppressing 
the corpuscles and imagining the ether traversed in 
all directions by luminous waves coming from all 
points of space. When a material object receives a 
luminous wave, this wave exercises upon it a mechani- 
cal action due to the Maxwell-Bartholi pressure, just as 
if it had received a blow from a material projectile. 
The waves in question may accordingly play the part 
of Lesage's corpuscles. This is admitted, for instance, 
by M. Tommasina. 

This does not get over the difficulties. The velocity 
of transmission cannot be greater than that of light, 
and we are thus brought to an inadmissible figure for 
the resistance of the medium. Moreover, if the light 
is wholly reflected, the effect is nil, just as in the 
hypothesis of the perfectly elastic corpuscles. In 
order to create attraction, the light must be partially 
absorbed, but in that case heat will be produced. The 
calculations do not differ essentially from those made 
in regard to Lesage's ordinary theory, and the result 
retains the same fantastic character. 

On the other hand, attraction is not absorbed, or 
but very slightly absorbed, by the bodies it traverses, 
while this is not true of the light we know. Light 
that would produce Newtonian attraction would re- 
quire to be very different from ordinary light, and 
to be, for instance, of very short wave length. This 
makes no allowance for the fact that, if our eyes were 
sensible to this light, the whole sky would appear 
much brighter than the Sun, so that the Sun would 
be seen to stand out in black, as otherwise it would 


repel instead of attracting us. For all these reasons, 
the light that would enable us to explain attraction 
would require to be much more akin to Rontgen's 
X rays than to ordinary light. 

Furthermore, the X rays will not do. However 
penetrating they may appear to us, they cannot pass 
through the whole Earth, and we must accordingly 
imagine X' rays much more penetrating than the 
ordinary X rays. Then a portion of the energy of 
these X' rays must be destroyed, as otherwise there 
would be no attraction. If we do not wish it to be 
transformed into heat, which would lead to the pro- 
duction of an enormous heat, we must admit that it 
is radiated in all directions in the form of secondary 
rays, which we may call X" rays, which must be much 
more penetrating even than the X' rays, failing which 
they would in their turn disturb the phenomena of 

Such are the complicated hypotheses to which we 
are led when we seek to make Lesage's theory tenable. 

But all that has been said assumes the ordinary 
laws of Mechanics. Will the case be stronger if we 
admit the new Dynamics ? And in the first place, can 
we preserve the Principle of Relativity ? First let us 
give Lesage's theory its original form, and imagine 
space furrowed by material corpuscles. If these 
corpuscles were perfectly elastic, the laws of their 
collision would be in conformity with this Principle 
of Relativity, but we know that in that case their effect 
would be nil. We must therefore suppose that these 
corpuscles are not elastic ; and then it is difficult to 
imagine a law of collision compatible with the Prin- 
ciple of Relativity. Besides, we should still get a 


considerable production of heat, and, notwithstanding 
that, a very appreciable resistance of the medium. 

If we suppress the corpuscles and return to the 
hypothesis of the Maxwell-Bartholi pressure, the 
difficulties are no smaller. It is this that tempted 
Lorentz himself in his Memoire to the Academy of 
Sciences of Amsterdam of the 25th of April 1900. 

Let us consider a system of electrons immersed in 
an ether traversed in all directions by luminous waves. 
One of these electrons struck by one of these waves 
will be set in vibration. Its vibration will be syn- 
chronous with that of the light, but there may be a 
difference of phase, if the electron absorbs a part ol 
the incident energy. If indeed it absorbs energy, it 
means that it is the vibration of the ether that keeps 
the electron in vibration, and the electron must ac- 
cordingly be behind the ether. An electron in motion 
may be likened to a convection current, therefore 
every magnetic field, and particularly that due to the 
luminous disturbance itself, must exercise a mechani- 
cal action upon the electron. This action is very 
slight, and more than that, it changes its sign in the 
course of the period ; nevertheless the mean action 
is not nil if there is a difference of phase between 
the vibrations of the electron and those of the ether 
The mean action is proportional to this difference, 
and consequently to the energy absorbed by the 

I cannot here enter into the details of the calcula- 
tions. I will merely state that the final result is an 
attraction between any two electrons varying in the 
inverse ratio of the square of the distance, and pro- 
portional to the energy absorbed by the two electrons. 


There cannot, therefore, be attraction without ab- 
sorption of light, and consequently without production 
of heat, and it is this that determined Lorentz to 
abandon this theory, which does not differ funda- 
mentally from the Lesage-Maxwell-Bartholi theory. 
He would have been still more alarmed if he had 
pushed the calculations to the end, for he would have 
found that the Earth's temperature must increase 10^' 
degrees a second. 



I have attempted to give in a few words as com- 
plete an idea as possible of these new doctrines ; I 
have tried to explain how they took birth, as other- 
wise the reader would have had cause to be alarmed 
by their boldness. The new theories are not yet 
demonstrated — they are still far from it, and rest 
merely upon an aggregation of probabilities suffi- 
ciently imposing to forbid our treating them with 
contempt. Further experiments will no doubt teach 
us what we must finally think of them. The root of 
the question is in Kaufmann's experiment and such 
as may be attempted in verification of it. 

In conclusion, may I be permitted to express a 
wish? Suppose that in a few years from now these 
theories are subjected to new tests and come out trium- 
phant, our secondary education will then run a great 
risk. Some teachers will no doubt wish to make room 
for the new theories. Novelties are so attractive, and 
it is so hard not to appear sufficiently advanced ! At 
least they will wish to open up prospects to the chil- 


dren, who will be warned, before they are taught the 
ordinary mechanics, that it has had its day, and that 
at most it was only good for such an old fogey as 
Laplace. Then they will never become familiar with 
the ordinary mechanics. 

Is it good to warn them that it is only approximate ? 
Certainly, but not till later on ; when they are steeped 
to the marrow in the old laws, when they have got 
into the way of thinking in them, and are no longer 
in danger of unlearning them, then they may safely 
be shown their limitations. 

It is with the ordinary mechanics that they have to 
live ; it is the only kind they will ever have to apply. 
Whatever be the progress of motoring, our cars will 
never attain the velocities at which its laws cease to 
be true. The other is only a luxury, and we must not 
think of luxury until there is no longer any risk of 
its being detrimental to what is necessary. 




The considerations I wish to develop here have so 
far attracted but little attention from astronomers. I 
have merely to quote an ingenious idea of Lord 
Kelvin's, which has opened to us a new field of re- 
search, but still remains to be followed up. Neither 
have I any original results to make known, and all 
that I can do is to give an idea of the problems that 
are presented, but that no one, up to this time, has 
made it his business to solve. 

Every one knows how a great number of modern 
physicists represent the constitution of gases. Gases 
are composed of an innumerable multitude of mole- 
cules which are animated with great velocities, and 
cross and re-cross each other in all directions. These 
molecules probably act at a distance one upon another, 
but this action decreases very rapidly with the distance, 
so that their trajectories remain apparently rectilineal, 
and only cease to be so when two molecules happen 
to pass sufificiently close to one another, in which case 
their mutual attraction or repulsion causes them to 
deviate to right or left. This is what is sometimes 
called a collision, but we must not understand this 


word collision in its ordinary sense ; it is not necessary 
that the two molecules should come into contact, but 
only that they should come near enough to each other 
for their mutual attraction to become perceptible. 
The laws of the deviation they undergo are the same 
as if there had been an actual collision. 

It seems at first that the orderless collisions of this 
innumerable dust can only engender an inextricable 
chaos before which the analyst must retire. But the 
law of great numbers, that supreme law of chance, 
comes to our assistance. In face of a semi-disorder 
we should be forced to despair, but in extreme disorder 
this statistical law re-establishes a kind of average or 
mean order in which the mind can find itself again. 
It is the study of this mean order that constitutes the 
kinetic theory of gases ; it shows us that the velocities 
of the molecules are equally distributed in all directions, 
that the amount of these velocities varies for the dif- 
ferent molecules, but that this very variation is subject 
to a law called Maxwell's law. This law teaches us 
how many molecules there are animated with such and 
such a velocity. As soon as a gas departs from this 
law, the mutual collisions of the molecules tend to 
bring it back promptly, by modifying the amount 
and direction of their velocities. Physicists have 
attempted, and not without success, to explain in this 
manner the experimental properties of gases — for 
instance, Mariotte's (or Boyle's) law. 

Consider now the Milky Way. Here also we see 
an innumerable dust, only the grains of this dust are 
no longer atoms but stars; these grains also move 
with great velocities, they act at a distance one upon 
another, but this action is so slight at great distances 


that their trajectories are rectiHneal ; nevertheless, from 
time to time, two of them may come near enough 
together to be deviated from their course, like a comet 
that passed too close to Jupiter. In a word, in the eyes 
of a giant, to whom our Suns were what our atoms 
are to us, the Milky Way would only look like a 
bubble of gas. 

Such was Lord Kelvin's leading idea. What can 
we draw from this comparison, and to what extent is 
it accurate ? This is what we are going to enquire 
into together ; but before arriving at a definite con- 
clusion, and without wishing to prejudice the question, 
we anticipate that the kinetic theory of gases will be, 
for the astronomer, a model which must not be 
followed blindly, but may afford him useful inspira- 
tion. So far celestial mechanics has attacked only 
the Solar System, or a few systems of double stars. 
It retired before the aggregations presented by the 
Milky Way, or clusters of stars, or resoluble nebulae, 
because it saw in them only chaos. But the Milky 
Way is no more complicated than a gas ; the statistical 
methods based upon the calculation of probabilities 
applicable to the one are also applicable to the other. 
Above all, it is important to realize the resemblance 
and also the difference between the two cases. 

Lord Kelvin attempted to determine by this means 
the dimensions of the Milky Way. For this purpose 
we are reduced to counting the stars visible in our 
telescopes, but we cannot be sure that, behind the 
stars we see, there are not others which we do not 
see ; so that what we should measure in this manner 
would not be the size of the Milky Way, but the scope 
of our instruments. The new theory will offer us other 


resources. We know, indeed, the motions of the stars 
nearest to us, and we can form an idea of the amount 
and direction of their velocities. If the ideas ex- 
pounded above are correct, these velocities must follow 
Maxwell's law, and their mean value will teach us, so 
to speak, what corresponds with the temperature of 
our fictitious gas. But this temperature itself depends 
upon the dimensions of our gaseous bubble. How, in 
fact, will a gaseous mass, left undisturbed in space, 
behave, if its elements are attracted in accordance 
with Newton's law? It will assume a spherical shape ; 
further, in consequence of gravitation, the density will 
be greater at the centre, and the pressure will also 
increase from the surface to the centre on account of 
the weight of the exterior parts attracted towards the 
centre ; lastly, the temperature will increase towards 
the centre, the temperature and the pressure being 
connected by what is called the adiabatic law, as is 
the case in the successive layers of our atmosphere. 
At the surface itself the pressure will be nil, and the 
same will be true of the absolute temperature, that is 
to say, of the velocity of the molecules. 

Here a question presents itself I have spoken of 
the adiabatic law, but this law is not the same for all 
gases, .since it depends upon the proportion of their 
two specific heats. For air and similar gases this pro- 
portion is 1.41 ; but is it to air that the Milky Way 
should be compared ? Evidently not. It should be 
regarded as a monatomic gas, such as mercury vapour, 
argon, or helium — that is to say, the proportion of the 
specific heats should be taken as equal to 1.66. And, 
indeed, one of our molecules would be, for instance, the 
Solar System ; but the planets are very unimportant 


personages and the Sun alone counts, so that our 
molecule is clearly monatomic. And even if we take 
a double star, it is probable that the action of a foreign 
star that happened to approach would become suffi- 
ciently appreciable to deflect the general motion of 
the system long before it was capable of disturbing 
the relative orbits of the two components. In a word, 
the double star would behave like an indivisible atom. 

However this may be, the pressure, and consequently 
the temperature, at the centre of the gaseous sphere 
are proportional to the size of the sphere, since the 
pressure is increased by the weight of all the over- 
lying strata. We may suppose that we are about at 
the centre of the Milky Way, and, by observing the 
actual mean velocity of the stars, we shall know what 
corresponds to the central temperature of our gaseous 
sphere and be able to determine its radius. 

We may form an idea of the result by the following 
considerations. Let us make a simple hypothesis. 
The Milky Way is spherical, and its masses are dis- 
tributed homogeneously : it follows that the stars 
describe ellipses having the same centre. If we sup- 
pose that the velocity drops to nothing at the surface, 
we can calculate this velocity at the centre by the 
equation of vis viva. We thus find that this velocity 
is proportional to the radius of the sphere and the 
square root of its density. If the mass of this sphere 
were that of the Sun, and its radius that of the ter- 
restrial orbit, this velocity, as is easily seen, would be 
that of the Earth upon its orbit. But in the case we 
have supposed, the Sun's mass would have to be 
distributed throughout a sphere with a radius 1,000,000 
times as great, this radius being the distance of the 

(1,777) 17 


nearest stars. The density is accordingly lO^' times 
as small ; now the velocities are upon the same scale, 
and therefore the radius must be lO* as great, or l,ooo 
times the distance of the nearest stars, which would 
give about a thousand million stars in the Milky Way. 

But you will tell me that these hypotheses are very 
far removed from reality. Firstly, the Milky Way is 
not spherical (we shall soon return to this point) ; and 
secondly, the kinetic theory of gases is not compatible 
with the hypothesis of a homogeneous sphere. But if 
we made an exact calculation in conformity with this 
theory, though we should no doubt obtain a different 
result, it would still be of the same order of magni- 
tude : now in such a problem the data are so uncertain 
that the order of magnitude is the only end we can 
aim at. 

And here a first observation suggests itself Lord 
Kelvin's result, which I have just obtained again by 
an approximate calculation, is in marked accordance 
with the estimates that observers have succeeded in 
making with their telescopes, so that we must conclude 
that we are on the point of piercing the Milky Way. 
But this enables us to solve another question. There 
are the stars we see because they shine, but might 
there not be dark stars travelling in the interstellar 
spaces, whose existence might long remain unknown? 
But in that case, what Lord Kelvin's method gives us 
would be the total number of stars, including the dark 
stars, and as his figure compares with that given by 
the telescope, there is not any dark matter, or at least 
not as much dark as there is brilliant matter. 

Before going further we must consider the problem 
under another aspect. Is the Milky Way, thus con- 


stituted, really the image of a gas properly so called ? 
We know that Crookes introduced the notion of a 
fourth state of matter, in which gases, becoming too 
rarefied, are no longer true gases, but become what he 
calls radiant matter. In view of the slightness of its 
density, is the Milky Way the image of gaseous or of 
radiant matter? It is the consideration of what is 
called Xh^ free path of the molecules that will supply 
the answer. 

A gaseous molecule's trajectory may be regarded 
as composed of rectilineal segments connected by 
very small arcs corresponding with the successive 
collisions. The length of each of these segments is 
what is called the free path. This length is obviously 
not the same for all the segments and for all the 
molecules ; but we may take an average, and this is 
called the mean free path, and its length is in inverse 
proportion to the density of the gas. Matter will be 
radiant when the mean path is greater than the 
dimensions of the vessel in which it is enclosed, so 
that a molecule is likely to traverse the whole vessel 
in which the gas is enclosed, without experiencing a 
collision, and it remains gaseous when the contrary 
is true. It follows that the same fluid may be radiant 
in a small vessel and gaseous in a large one, and this 
is perhaps the reason why, in the case of Crookes' 
tubes, a more perfect vacuum is required for a larger 

What, then, is the case of the Milky Way? It is 
a mass of gas of very low density, but of very great 
dimensions. Is it likely that a star will traverse it 
without meeting with any collision — that is to say, 
without passing near enough to another star to be 


appreciably diverted from its course? What do we 
mean by near enough ? This is necessarily somewhat 
arbitrary, but let us assume that it is the distance 
from the Sun to Neptune, which represents a deviation 
of about ten degrees. Supposing, now, that each of 
our stars is surrounded by a danger sphere of this 
radius, will a straight line be able to pass between 
these spheres ? At the mean distance of the stars of 
the Milky Way, the radius of these spheres will sub- 
tend an angle of about a tenth of a second, and we 
have a thousand million stars. If we place upon the 
celestial sphere a thousand million little circles with 
radius of a tenth of a second, will these circles cover 
the celestial sphere many times over? Far from it. 
They will only cover a sixteen-thousandth part. Thus 
the Milky Way is not the image of gaseous matter, 
but of Crookes' radiant matter. Nevertheless, as there 
was very little precision in our previous conclusions, 
we do not require to modify them to any appreciable 

But there is another difficulty. The Milky Way is 
not spherical, and up to now we have reasoned as 
though it were so, since that is the form of equilibrium 
that would be assumed by a gas isolated in space. 
On the other hand, there are clusters of stars whose 
form is globular, to which what we have said up to 
this point would apply better. Herschel had already 
applied himself to the explanation of their remarkable 
appearance. He assumed that the stars of these 
clusters are uniformly distributed in such a way that 
a cluster is a homogeneous sphere. Each star would 
then describe an ellipse, and all these orbits would be 
accomplished in the same time, so that at the end of 


a certain period the cluster would return to its original 
configuration, and that configuration would be stable. 
Unfortunately the clusters do not appear homogene- 
ous. We observe a condensation at the centre, and 
we should still observe it even though the sphere were 
homogeneous, since it is thicker at the centre, but it 
would not be so marked. A cluster may, therefore, 
better be compared to a gas in adiabatic equilibrium 
which assumes a spherical form, because that is the 
figure of equilibrium of a gaseous mass. 

But, you will say, these clusters are much smaller 
than the Milky Way, of which it is even probable that 
they form a part, and although they are denser, they 
give us rather something analogous to radiant matter. 
Now, gases only arrive at their adiabatic equilibrium 
in consequence of innumerable collisions of the mole- 
cules. We might perhaps find a method of reconciling 
these facts. Suppose the stars of the cluster have just 
sufficient energy for their velocity to become nil when 
they reach the surface. Then they may traverse the 
cluster without a collision, but on reaching the surface 
they turn back and traverse it again. After traversing 
it a great number of times, they end by being deflected 
by a collision. Under these conditions we should still 
have a matter that might be regarded as gaseous. If 
by chance there were stars in the cluster with greater 
velocities, they have long since emerged from it, and 
have left it never to return. For all these reasons it 
would be interesting to examine the known clusters 
and try to get an idea of the law of their densities and 
see if it is the adiabatic law of gases. 

But to return to the Milky Way. It is not spherical, 
and would be more properly represented as a flattened 


disc. It is clear, then, that a mass starting without 
velocity from the surface will arrive at the centre with 
varying velocities, according as it has started from the 
surface in the neighbourhood of the middle of the disc 
or from the edge of the disc. In the latter case the 
velocity will be considerably greater. 

Now up to the present we have assumed that the 
individual velocities of the stars, the velocities we 
observe, must be comparable to those that would be 
attained by such masses. This involves a certain 
difficulty. I have given above a value for the dimen- 
sions of the Milky Way, and I deduced it from the 
observed individual velocities, which are of the same 
order of magnitude as that of the Earth upon its orbit ; 
but what is the dimension I have thus measured ? Is 
it the thickness or the radius of the disc ? It is, no 
doubt, something between the two, but in that case 
what can be said of the thickness itself, or of the 
radius of the disc ? Data for making the calculation 
are wanting, and I content myself with foreshadowing 
the possibility of basing at least an approximate 
estimate upon a profound study of the individual 

Now, we find ourselves confronted by two hypo- 
theses. Either the stars of the Milky Way are 
animated with velocities which are in the main 
parallel with the Galactic plane, but otherwise dis- 
tributed uniformly in all directions parallel with 
this plane. If so, observation of the individual 
motions should reveal a preponderance of components 
parallel with the Milky Way. This remains to be 
ascertained, for I do not know that any systematic 
study has been made from this point of view. On the 


other hand, such an equiHbrium could only be pro- 
visional, for, in consequence of collisions, the molecules 
— I mean the stars — will acquire considerable velocities 
in a direction perpendicular to the Milky Way, and 
will end by emerging from its plane, so that the 
system will tend towards the spherical form, the only 
figure of equilibrium of an isolated gaseous mass. 

Or else the whole system is animated with a common 
rotation, and it is for this reason that it is flattened, 
like the Earth, like Jupiter, and like all rotating 
bodies. Only, as the flattening is considerable, the 
rotation must be rapid. Rapid, no doubt, but we 
must understand the meaning of the word. The 
density of the Milky Way is 10^^ times as low as the 
Sun's ; a velocity of revolution v'lo'-^ times smaller 
than the Sun's would therefore be equivalent in its 
case from the point of view of the flattening. A 
velocity 10^- times as slow as the Earth's, or the 
thirtieth of a second of arc in a century, will be a 
very rapid revolution, almost too rapid for stable 
equilibrium to be possible. 

In this hypothesis, the observable individual motions 
will appear to us uniformly distributed, and there will 
be no more preponderance of the components parallel 
with the Galactic plane. They will teach us nothing 
with respect to the rotation itself, since we form part 
of the rotating system. If the spiral nebulae are other 
Milky Ways foreign to ours, they are not involved 
in this rotation, and we might study their individual 
motions. It is true that they are very remote, for if 
a nebula has the dimensions of the Milky Way, and 
if its apparent radius is, for instance, 20", its distance 
is 10,000 times the radius of the Milky Way. 


But this does not matter, since it is not about the 
rectilinear motion of our system that we ask them for 
information, but about its rotation. The fixed stars, 
by their apparent motion, disclose the diurnal rotation 
of the Earth, although their distance is immense. 
Unfortunately, the possible rotation of the Milky 
Way, rapid as it is, relatively speaking, is very slow 
from the absolute point of view, and, moreover, bear- 
ings upon nebula; cannot be very exact. It would 
accordingly require thousands of years of observation 
to learn anything. 

However it be, in this second hypothesis, the figure 
of the Milky Way would be a figure of ultimate 

I will not discuss the relative value of these two 
hypotheses at any greater length, because there is a 
third which is perhaps more probable. We know that 
among the irresoluble nebulae several families can be 
distinguished, the irregular nebulae such as that in 
Orion, the planetary and annular nebulae, and the 
spiral nebulae. The spectra of the first two families 
have been determined, and prove to be discontinuous. 
These nebulae are accordingly not composed of stars. 
Moreover, their distribution in the sky appears to 
depend upon the Milky Way, whether they show a 
tendency to be removed from it, or on the contrary 
to approach it, and therefore they form part of the 
system. On the contrary, the spiral nebulae are 
generally considered as independent of the Milky 
Way : it is assumed that they are, like it, composed 
of a multitude of stars ; that they are, in a word, 
other Milky Ways very remote from ours. The work 
recently done by Stratonoff tends to make us look 


upon the Milky Way itself as a spiral nebula, and this 
is the third hypothesis of which I wished to speak. 

How are we to explain the very singular appear- 
ances presented by the spiral nebula, which are too 
regular and too constant to be due to chance? To 
begin with, it is sufficient to cast one's eyes upon one 
of these figures to see that the mass is in rotation, and 
we can even see the direction of the rotation : all the 
spiral radii are curved in the same direction, and it is 
evident that it is the advancing wing hanging back 
upon the pivot, and that determines the direction of 
the rotation. But that is not all. It is clear that 
these nebulae cannot be likened to a gas in repose, 
nor even to a gas in relative equilibrium under the 
domination of a uniform rotation ; they must be 
compared to a gas in permanent motion in which 
internal currents rule. 

Suppose, for example, that the rotation of the central 
nucleus is rapid (you know what I mean by this word), 
too rapid for stable equilibrium. Then at the equator 
the centrifugal force will prevail over the attraction, 
and the stars will tend to escape from the equator, 
and will form divergent currents. But as they recede, 
since their momentum ot rotation remains constant 
and the radius vector increases, their angular velocity 
will diminish, and it is for this reason that the advan- 
cing wing appears to hang back. 

Under this aspect of the case there would not be 
a true permanent motion, for the central nucleus 
would constantly lose matter which would go out 
never to return, and would be gradually exhausted. 
But we may modify the hypothesis. As it recedes, 
the star loses its velocity and finally stops. At that 


moment the attraction takes possession of it again and 
brings it back towards the nucleus, and accordingly 
there will be centripetal currents. We must assume 
that the centripetal currents are in the first rank and 
the centrifugal currents in the second rank, if we take 
as a comparison a company in battle executing a 
turning movement. Indeed the centrifugal force must 
be compensated by the attraction exercised by the 
central layers of the swarm upon the exterior layers. 

Moreover, at the end of a certain length of time, 
a permanent status is established. As the swarm 
becomes curved, the attraction exercised by the 
advancing wing upon the pivot tends to retard the 
pivot, and that of the pivot upon the advancing wing 
tends to accelerate the advance of this wing, whose 
retrograde motion increases no further, so that finally 
all the radii end by revolving at a uniform velocity. 
We may nevertheless assume that the rotation of the 
nucleus is more rapid than that of the radii. 

One question remains. Why do these centripetal 
and centrifugal swarms tend to concentrate into radii 
instead of being dispersed more or less throughout, 
and why are these radii regularly distributed ? The 
reason for the concentration of the swarms is the 
attraction exercised by the swarms already existing 
upon the stars that emerge from the nucleus in their 
neighbourhood. As soon as an inequality is produced, 
it tends to be accentuated by this cause. 

Why are the radii regularly distributed ? This is 
a more delicate matter. Suppose there is no rotation, 
and that all the stars are in two rectangular planes in 
such a way that their distribution is symmetrical in 
relation to the two planes. By symmetry, there would 


be no reason for their emerging from the planes nor 
for the symmetry to be altered. This configuration 
would accordingly give equilibrium, but it would be an 
unstable equilibrium. 

If there is rotation on the contrary, we shall get 
an analogous configuration of equilibrium with four 
curved radii, equal to one another, and intersecting at 
an angle of 90°, and if the rotation is sufficiently 
rapid, this equilibrium may be stable. 

I am not in a position to speak more precisely. It 
is enough for me to foreshadow the possibility that 
these spiral forms may, perhaps, some day be ex- 
plained by the help only of the law of gravitation and 
statistical considerations, recalling those of the theory 
of gases. 

What I have just said about internal currents shows 
that there might be some interest in a systematic 
study of the aggregate of the individual motions. 
This might be undertaken a hundred years hence, 
when the second edition of the astrographic chart of 
the heavens is brought out and compared with the 
first, the one that is being prepared at present. 

But I should wish, in conclusion, to call your 
attention to the question of the age of the Milky Way 
and the nebulae. We might form an idea of this age 
if we obtained confirmation of what we have imagined 
to be the case. This kind of statistical equilibrium of 
which gases supply the model, cannot be established 
except as a consequence of a great number of col- 
lisions. If these collisions are rare, it can only be 
produced after a very long time. If actually the 
Milky Way (or at least the clusters that form par<- 
of it), and if the nebulae have obtained this equilibrium, 


it is because they are very ancient, and we shall get an 
inferior limit for their age. We shall likewise obtain a 
superior limit, for this equilibrium is not ultimate and 
cannot last for ever. Our spiral nebula; would be com- 
parable to gases animated with permanent motions. 
But gases in motion are viscous and their velocities 
are finally expended. What corresponds in this case 
to viscidity (and depends upon the chances of collision 
of the molecules) is exceedingly slight, so that the 
actual status may continue for a very long time, but 
not for ever, so that our Milky Ways cannot be ever- 
lasting nor become infinitely ancient. 

But this is not all. Consider our atmosphere. At 
the surface an infinitely low temperature must prevail, 
and the velocity of the molecules is in the neighbour- 
hood of zero. But this applies only to the mean 
velocity. In consequence of collisions, one of these 
molecules may acquire (rarely, it is true) an enormous 
velocity, and then it will leave the atmosphere, and 
once it has left it, it will never return. Accordingly 
our atmosphere is being exhausted exceedingly slowly. 
By the same mechanism the Milky Way will also lose 
a star from time to time, and this likewise limits its 

Well, it is certain that if we calculate the age of 
the Milky Way by this method, we shall arrive at 
enormous figures. But here a difficulty presents itself 
Certain physicists, basing their calculations on other 
considerations, estimate that Suns can have but an ephe- 
meral existence of about fifty millions of years, while 
our minimum would be much greater than that. Must 
we believe that the evolution of the Milky Way began 
while matter was still dark? But how have all the 


stars that compose it arrived at the same time at the 
adult period, a period which lasts for so short a time ? 
Or do they all reach it successively, and are those that 
we see only a small minority as compared with those 
that are extinct or will become luminous some day ? 
But how can we reconcile this with what has been said 
above about the absence of dark matter in any con- 
siderable proportion ? Must we abandon one of the 
two hypotheses, and, if so, which ? I content myself 
with noting the difficulty, without pretending to solve 
it, and so I end with a great mark of interrogation. 
Still, it is interesting to state problems even though 
their solution seems very remote. 



Every one understands what an interest we have in 
knowing- the shape and the dimensions of our globe, 
but some people would perhaps be astonished at the 
precision that is sought for. Is this a useless luxury ? 
What is the use of the efforts geodesists devote to it ? 

If a Member of Parliament were asked this question, 
I imagine he would answer : " I am led to think that 
Geodesy is one of the most useful of sciences, for it is 
one of those that cost us most money." I shall 
attempt to give a somewhat more precise answer. 

The great works of art, those of peace as well as 
those of war, cannot be undertaken without long 
studies, which save many gropings, miscalculations, 
and useless expense. These studies cannot be made 
without a good map. But a map is nothing but a 
fanciful picture, of no value whatever if we try to 
construct it without basing it upon a solid framework. 
As well might we try to make a human body stand 
upright with the skeleton removed. 

Now this framework is obtained by geodetic meas- 

* Throughout this chapter the author is speaking of the work of his 
own countrymen. In the translation such words as "we" and "our" 
have been avoided, as far as possible ; but where they occur, they must 
be understood to refer to P'rance and not to England. 


urements. Therefore without Geodesy we can have 
no good map, and without a good map no great 
public works. 

These reasons would no doubt be sufficient to justify 
much expense, but they are reasons calculated to con- 
vince practical men. It is not upon these that we 
should insist here ; there are higher and, upon the 
whole, more important reasons. 

We will therefore state the question differently : 
Can Geodesy make us better acquainted with nature ? 
Does it make us understand its unity and harmony? 
An isolated fact indeed is but of little worth, and the 
conquests of science have a value only if they prepare 
new ones. 

Accordingly, if we happened to discover a little 
hump upon the terrestrial ellipsoid, this discovery 
would be of no great interest in itself It would 
become precious on the contrary if, in seeking for the 
cause of the hump, we had the hope of penetrating 
new secrets. 

So when Maupertuis and La Condamine in the 
eighteenth century braved such diverse climates, it 
was not only for the sake of knowing the shape of our 
planet, it was a question of the system of the whole 
World. If the Earth was flattened, Newton was 
victorious, and with him the doctrine of gravitation 
and the whole of the modern celestial mechanics. 
And to-day, a century and a half since the victory 
of the Newtonians, are we to suppose that Geodesy 
has nothing more to teach us ? We do not know 
what there is in the interior of the globe. Mine 
shafts and borings have given us some knowledge 
of a stratum one or two miles deep — that is to say, 


the thousandth part of the total mass ; but what is 
there below that? 

Of all the extraordinary voyages dreamed of by 
Jules Verne, it was perhaps the voyage to the centre of 
the Earth that led us to the most unexplored regions. 

But those deep sunk rocks that we cannot reach, 
exercise at a distance the attraction that acts upon 
the pendulum and deforms the terrestrial spheroid. 
Geodesy can therefore weigh them at a distance, so to 
speak, and give us information about their disposition. 
It will thus enable us really to see those mysterious 
regions which Jules Verne showed us only in imagi- 

This is not an empty dream. By comparing all the 
measurements, M. Faye has reached a result well 
calculated to cause surprise. In the depths beneath 
the oceans, there are rocks of very great density, while, 
on the contrary, beneath the continents there seem 
to be empty spaces. 

New observations will perhaps modify these con- 
clusions in their details, but our revered master has, at 
any rate, shown us in what direction we must push 
our researches, and what it is that the geodesist can 
teach the geologist who is curious about the interior 
constitution of the Earth, and what material he can 
supply to the thinker who wishes to reflect upon the 
past and the origin of this planet. 

Now why have I headed this chapter French 
Geodesy? It is because, in different countries, this 
science has assumed, more perhaps than any other, 
a national character ; and it is easy so see the reason 
for this. 

There must certainly be rivalries. Scientific rivalries 


are always courteous, or, at least, almost always. In 
any case they are necessary, because they are always 

Well, in these enterprises that demand such long 
efforts and so many collaborators, the individual is 
effaced, in spite of himself of course. None has the 
right to say, this is my work. So the rivalry is not 
between individuals, but between nations. Thus we 
are led to ask what share France has taken in the 
work, and I think we have a right to be proud of 
what she has done. 

At the beginning of the eighteenth century there 
arose long discussions between the Newtonians, who 
believed the Earth to be flattened as the theory of 
gravitation demands, and Cassini, who was misled b)' 
inaccurate measurements, and believed the globe to 
be elongated. Direct observation alone could settle 
the question. It was the French Academy of Sciences 
that undertook this task, a gigantic one for that 

While Maupertuis and Clairaut were measuring a 
degree of longitude within the Arctic circle, Bouguer 
and La Condamine turned their faces towards the 
mountains of the Andes, in regions that were then 
subject to Spain, and to-day form the Republic of 
Ecuador. Our emissaries were exposed to great 
fatigues, for journeys then were not so easy as they 
are to-day. 

It is true that the country in which Maupertuis' 

operations were conducted was not a desert, and it is 

even said that he enjoyed among the Lapps those soft 

creature comforts that are unknown to the true Arctic 

navigator. It was more or less in the neighbourhood 
(1,777) 18 


of places to which, in our day, comfortable steamers 
carry, every summer, crowds of tourists and young 
English ladies. But at that date Cook's Agency did 
not exist, and Maupertuis honestly thought that he 
had made a Polar expedition. 

Perhaps he was not altogether wrong. Russians 
and Swedes are to-day making similar measurements 
at Spitzbergen, in a country where there are real ice- 
packs. But their resources are far greater, and the 
difference of date fully compensates for the difference 
of latitude. 

Maupertuis' name has come down to us considerably 
mauled by the claws of Dr. Akakia, for Maupertuis 
had the misfortune to displease Voltaire, who was 
then king of the mind. At first he was extravagantly 
praised by Voltaire ; but the flattery of kings is as 
much to be dreaded as their disfavour, for it is followed 
by a terrible day of reckoning. Voltaire himself learnt 
something of this. 

Voltaire called Maupertuis " my kind master of 
thought," " Marquess of the Arctic Circle," " dear 
flattener of the world and of Cassini," and even, as 
supreme flattery, " Sir Isaac Maupertuis " ; and he 
wrote, " There is none but the King of Prussia that 
I place on a level with you ; his sole defect is that he 
is not a geometrician." But very soon the scene 
changes ; he no longer speaks of deifying him, like 
the Argonauts of old, or of bringing down the council 
of the gods from Olympus to contemplate his work, 
but of shutting him up in a mad-house. He speaks 
no more of his sublime mind, but of his despotic pride, 
backed by very little science and much absurdity. 

1 do not wish to tell the tale of these mock-heroic 


conflicts, but I should like to make a few reflections 
upon two lines of Voltaire's. In his Discours sur la 
Moderation (there is no question of moderation in 
praise or blame), the poet wrote :■ — 

Vous avez confirme dans des lieux pleins d 'ennui 
Ce que Newton connut sans sortir de chez lui. 

(You have confirmed, in dreary far-off lands, 
What Newton knew without e'er leaving home.) 

These two lines, which take the place of the hyper- 
bolical praises of earlier date, are most unjust, and 
without any doubt, Voltaire was too well informed 
not to realize it. 

At that time men valued only the discoveries that 
can be made without leaving home. To-day it is 
theory rather that is held in low esteem. But this 
implies a misconception of the aim of science. 

Is nature governed by caprice, or is harmony the 
reigning influence ? That is the question. It is when 
science reveals this harmony that it becomes beauti- 
ful, and for that reason worthy of being cultivated. 
But whence can this revelation come if not from the 
accordance of a theory with experience ? Our aim 
then is to find out whether or not this accordance 
exists. From that moment, these two terms, which 
must be compared with each other, become one as 
indispensable as the other. To neglect one for the 
other would be folly. Isolated, theory is empty and 
experience blind ; and both are useless and of no 
interest alone. 

Maupertuis is therefore entitled to his share of the 
fame. Certainly it is not equal to that of Newton, 
who had received the divine spark, or even of his 


collaborator Clairaut It is not to be despised, how- 
ever, because his work was necessary ; and if France, 
after being outstripped by England in the seventeenth 
century, took such full revenge in the following cen- 
tury, it was not only to the genius of the Clairauts, 
the d'Alemberts, and the Laplaces that she owed 
it, but also to the long patience of such men as 
Maupertuis and La Condamine. 

We come now to what may be called the second 
heroic period of Geodesy. France was torn with 
internal strife, and the whole of Europe was in arms 
against her. One would suppose that these tre- 
mendous struggles must have absorbed all her ener- 
gies. Far from that, however, she had still some left 
for the service of science. The men of that day 
shrank before no enterprise — they were men of faith. 

Delambre and M^chain were commissioned to 
measure an arc running from Dunkirk to Barcelona. 
This time there is no journey to Lapland or Peru ; 
the enemy's squadrons would close the roads. But 
if the expeditions are less distant, the times are so 
troublous that the obstacles and even the dangers 
are quite as great. 

In France Delambre had to fight against the ill- 
will of suspicious municipalities. One knows that 
steeples, which can be seen a long way off, and ob- 
served with precision, often serve as signals for 
geodesists. But in the country Delambre was working 
through, there were no steeples left. I forget now 
what proconsul it was who had passed through it and 
boasted that he had brought down all the steeples 
that raised their heads arrogantly above the humble 
dwellings of the common people. 


So they erected pyramids of planks covered with 
white linen to make them more conspicuous. This 
was taken to mean something quite different. White 
Hnen ! Who was the foolhardy man who ventured 
to set up, on our heights .so recently liberated, the 
odious standard of the counter-revolution ? The 
white linen must needs be edged with blue and red 

Mechain, operating in Spain, met with other but 
no less serious difficulties. The Spanish country 
folk were hostile. There was no lack of steeples, 
but was it not sacrilege to take possession of them 
with instruments that were mysterious and perhaps 
diabolical ? The revolutionaries were the allies of 
Spain, but they were allies who smelt a little of the 

"We are constantly threatened," writes Mechain, 
"with having our throats cut." Happily, thanks to 
the exhortations of the priests, and to the pastoral 
letters from the bishops, the fiery Spaniards con- 
tented themselves with threats. 

Some years later, Mdchain made a second expedi- 
tion to Spain. He proposed to extend the meridian 
from Barcelona to the Balearic Isles. This was the 
first time that an attempt had been made to cross a 
large arm of the sea by triangulation, by taking 
observations of signals erected upon some high moun- 
tain in a distant island. The enterprise was well 
conceived and well planned, but it failed nevertheless. 
The French scientist met with all kinds of difficulties, 
of which he complains bitterly in his correspondence. 
" Hell," he writes, perhaps with some exaggeration, 
" hell, and all the scourges it vomits upon the earth— 


storms, war, pestilence, and dark intrigues — are let 
loose against me ! " 

The fact is that he found among his collaborators 
more headstrong arrogance than good-will, and that 
a thousand incidents delayed his work. The plague 
was nothing ; fear of the plague was much more 
formidable. All the islands mistrusted the neighbour- 
ing islands, and were afraid of receiving the scourge 
from them. It was only after long weeks that 
M^chain obtained permission to land, on condition of 
having all his papers vinegared — such were the anti- 
septics of those days. Disheartened and ill, he had 
just applied for his recall, when he died. 

It was Arago and Biot who had the honour of 
taking up the unfinished work and bringing it to a 
happy conclusion. Thanks to the support of the 
Spanish Government and the protection of several 
bishops, and especially of a celebrated brigand chief, 
the operations progressed rapidly enough. They were 
happily terminated, and Biot had returned to France, 
when the storm burst. 

It was the moment when the whole of Spain was 
taking up arms to defend her independence against 
France. Why was this stranger climbing mountains 
to make signals ? It was evidently to call the French 
army. Arago only succeeded in escaping from the 
populace by giving himself up as a prisoner. In his 
prison his only distraction was reading the account 
of his own execution in the Spanish newspapers. The 
newspapers of those days sometimes gave premature 
news. He had at least the consolation of learning 
that he had died a courageous and a Christian death. 

Prison itself was not safe, and he had to make his 


escape and reach Algiers. Thence he sailed for Mar- 
seilles on an Algerian ship. This ship was captured 
by a Spanish privateer, and so Arago was brought 
back to Spain, and dragged from dungeon to dun- 
geon in the midst of vermin and in the most horrible 

If it had only been a question of his subjects and 
his guests, the Dey would have said nothing. But 
there were two lions on board, a present the African 
sovereign was sending to Napoleon. The Dey 
threatened war. 

The vessel and the prisoners were released. The 
point should have been correctly made, since there was 
an astronomer on board ; but the astronomer was sea- 
sick, and the Algerian sailors, who wished to go to 
Marseilles, put in at Bougie. Thence Arago travelled 
to Algiers, crossing Kabylia on foot through a thousand 
dangers. He was detained for a long time in Africa 
and threatened with penal servitude. At last he was 
able to return to France. His observations, which he 
had preserved under his shirt, and more extraordinary 
still, his instruments, had come through these terrible 
adventures without damage. 

Up to this point, France not only occupied the first 
place, but she held the field almost alone. In the 
years that followed she did not remain inactive, and 
the French ordnance map is a model. Yet the new 
methods of observation and of calculation came 
principally from Germany and England. It is only 
during the last forty years that France has regained 
her position. 

She owes it to a scientific officer, General Perrier, 
who carried out successfully a truly audacious enter- 


prise, the junction of Spain and Africa. Stations were 
established upon four peaks on the two shores of the 
Mediterranean. There were long months of waiting 
for a calm and clear atmosphere. At last there was 
seen the slender thread of light that had travelled 
two hundred miles over the sea, and the operation had 

To-day still more daring projects have been con- 
ceived. From a mountain in the vicinity of Nice 
signals are to be sent to Corsica, no longer with a 
view to the determination of geodetic questions, but 
in order to measure the velocity of light. The dis- 
tance is only one hundred and twenty-five miles, but 
the ray of light is to make the return journey, after 
being reflected from a mirror in Corsica. And it must 
not go astray on the journey, but must return to 
the exact spot from which it started. 

Latterly the activity of French Geodesy has not 
slackened. We have no more such astonishing 
adventures to relate, but the scientific work accom- 
plished is enormous. The territory of France beyond 
the seas, just as that of the mother country, is being 
covered with triangles measured with precision. 

We have become more and more exacting, and 
what was admired by our fathers does not satisfy 
us to-day. But as we seek greater exactness, the 
difficulties increase considerably. We are surrounded 
by traps, and have to beware of a thousand unsuspected 
causes of error. It becomes necessary to make more 
and more infallible instruments. 

Here again France has not allowed herself to be 
outdone. Her apparatus for the measurement of bases 
and of angles leaves nothing to be desired, and I would 


also mention Colonel Defforges' pendulum, which 
makes it possible to determine gravity with a pre- 
cision unknown till now. 

The future of French Geodesy is now in the hands 
of the geographical department of the army, which 
has been directed successively by General Bassot and 
General Berthaut. This has advantages that can 
hardly be overestimated. For good geodetic work, 
scientific aptitude alone is not sufficient. A man 
must be able to endure long fatigues in all climates. 
The chief must know how to command the obedience 
of his collaborators and to enforce it upon his native 
helpers. These are military qualities, and, moreover, it 
is known that science has always gone hand in hand 
with courage in the French army. 

I would add that a military organization assures 
the indispensable unity of action. It would be more 
difficult to reconcile the pretensions of rival scientists, 
jealous of their independence and anxious about what 
they call their honour, who would nevertheless have 
to operate in concert, though separated by great 
distances. There arose frequent discussions between 
geodesists of former times, some of which started 
echoes that were heard long after. The Academy 
long rang with the quarrel between Bouguer and 
La Condamine. I do not mean to say that soldiers 
are free from passions, but discipline imposes silence 
upon over-sensitive vanity. 

Several foreign governments have appealed to 
French officers to organize their geodetic depart- 
ments. This is a proof that the scientific influence of 
France abroad has not been weakened. 

Her hydrographic engineers also supply a famous 


contingent to the common work. The chart of her 
coasts and of her colonies, and the study of tides, offer 
them a vast field for research. Finally, I would 
mention the general levelling of France, which is 
being carried out by M. Lallemand's ingenious and 
accurate methods. 

With such men, we are sure of the future. Work for 
them to do will not be wanting. The French colonial 
empire offers them immense tracts imperfectly explored. 
And that is not all. The International Geodetic Asso- 
ciation has recognized the necessity of a new measure- 
ment of the arc of Quito, formerly determined by La 
Condamine. It is the French who have been entrusted 
with the operation. They had every right, as it was 
their ancestors who achieved, so to speak, the scientific 
conquest of the Cordilleras. Moreover, these rights 
were not contested, and the French Government 
determined to exercise them. 

Captains Maurain and Lacombe made a preliminary 
survey, and the rapidity with which they accomplished 
their mission, travelling through difficult countries, and 
climbing the most precipitous peaks, deserves the 
highest praise. It excited the admiration of General 
Alfaro, President of the Republic of Ecuador, who 
surnamed ^them los hombres de hierro, the men of 

The definitive mission started forthwith, under the 
command of Lieutenant-Colonel (then Commandant) 
Bourgeois. The results obtained justified the hopes 
that had been entertained. But the officers met with 
unexpected difficulties due to the climate. More than 
once one of them had to remain for several months at 
an altitude of 13,000 feet, in clouds and snow, without 


seeing anything of the signals he had to observe, which 
refused to show themselves. But thanks to their per- 
severance and courage, the only result was a delay, 
and an increase in the expenses, and the accuracy of 
the measurements did not suffer. 


What I have attempted to explain in the foregoing 
pages is how the scientist is to set about making a 
selection of the innumerable facts that are offered to 
his curiosity, since he is compelled to make a selection, 
if only by the natural infirmity of his mind, though a 
selection is always a sacrifice. To begin with, I ex- 
plained it by general considerations, recalling, on the 
one hand, the nature of the problem to be solved, and 
on the other, seeking a better understanding of the 
nature of the human mind, the principal instrument in 
the solution. Then I explained it by examples, but 
not an infinity of examples, for I too had to make 
a selection, and I naturally selected the questions 
I had studied most carefully. Others would no 
doubt have made a different selection, but this matters 
little, for I think they would have reached the same 

There is a hierarchy of facts. Some are without 
any positive bearing, and teach us nothing but them- 
selves. The scientist who ascertains them learns 
nothing but facts, and becomes no better able to 
foresee new facts. Such facts, it seems, occur but 
once, and are not destined to be repeated. 

There are, on the other hand, facts that give a large 


return, each of which teaches us a new law. And 
since he is obliged to make a selection, it is to these 
latter facts that the scientist must devote himself. 

No doubt this classification is relative, and arises 
from the frailty of our mind. The facts that give but 
a small return are the complex facts, upon which a 
multiplicity of circumstances exercise an appreciable 
influence — circumstances so numerous and so diverse 
that we cannot distinguish them all. But I should 
say, rather, that they are the facts that we consider 
complex, because the entanglement of these circum- 
stances exceeds the compass of our mind. No doubt 
a vaster and a keener mind than ours would judge 
otherwise. But that matters little ; it is not this 
superior mind that we have to use, but our own. 

The facts that give a large return are those that we 
consider simple, whether they are so in reality, because 
they are only influenced by a small number of well- 
defined circumstances, or whether they take on an 
appearance of simplicity, because the multiplicity of 
circumstances upon which they depend obey the laws 
of chance, and so arrive at a mutual compensation. 
This is most frequently the case, and is what com- 
pelled us to enquire somewhat closely into the 
nature of chance. The facts to which the laws of 
chance apply become accessible to the scientist, who 
would lose heart in face of the extraordinary com- 
plication of the problems to which these laws are not 

We have seen how these considerations apply not 
only to the physical but also to the mathematical 
sciences. The method of demonstration is not the 
same for the physicist as for the mathematician. But 


their methods of discovery are very similar. In the 
case of both they consist in rising from the fact to the 
law, and in seeking the facts that are capable of 
leading up to a law. 

In order to elucidate this point, I have exhibited 
the mathematician's mind at work, and that under 
three forms : the mind of the inventive and creative 
mathematician ; the mind of the unconscious geome- 
trician who, in the days of our far-off ancestors or in 
the hazy years of our infancy, constructed for us our 
instinctive notion of space ; and the mind of the youth 
in a secondary school for whom the master unfolds the 
first principles of the science, and seeks to make him 
understand its fundamental definitions. Through- 
out we have seen the part played by intuition and 
the spirit of generalization, without which these 
three grades of mathematicians, if I may venture 
so to express myself, would be reduced to equal 

And in demonstration itself logic is not all. The 
true mathematical reasoning is a real induction, 
differing in many respects from physical induction, 
but, like it, (proceeding from the particular to the 
universal. All the efforts that have been made to 
upset this order, and to reduce mathematical induction 
to the rules of logic, have ended in failure, but poorly 
disguised by the use of a language inaccessible to the 

The examples I have drawn from the physical 
sciences have shown us a good variety of instances of 
facts that give a large return. A single experiment of 
Kaufmann's upon radium rays revolutionizes at once 
Mechanics, Optics, and Astronomy. Why is this? It 


is because, as these sciences developed, we have recog- 
nized more clearly the links which unite them, and 
at last we have perceived a kind of general design of 
the map of universal science. There are facts com- 
mon to several sciences, like the common fountain 
head of streams diverging in all directions, which may- 
be compared to that nodal point of the St. Gothard 
from which there flow waters that feed four different 

Then we can make our selection of facts with more 
discernment than our predecessors, who regarded 
these basins as distinct and separated by impassable 

It is always simple facts that we must select, but 
among these simple facts we should prefer those that 
are situated in these kinds of nodal points of which 
I have just spoken. 

And when sciences have no direct link, they can 
still be elucidated mutually by analogy. When the 
laws that regulate gases were being studied, it was 
realized that the fact in hand was one that would give 
a great return, and yet this return was still estimated 
below its true value, since gases are, from a certain 
point of view, the image of the Milky Way ; and these 
facts, which seemed to be of interest only to the 
physicist, will soon open up new horizons to the 
astronomer, who little expected it. 

Lastly, when the geodesist finds that he has to turn 
his glass a few seconds of arc in order to point it upon 
a signal that he has erected with much difficulty, it is 
a very small fact, but it is a fact giving a great return, 
not only because it reveals the existence of a little 
hump upon the terrestrial geoid, for the little hump 


would of itself be of small interest, but because this 
hump gives him indications as to the distribution of 
matter in the interior of the globe, and, through that, 
as to the past of our planet, its future, and the laws of 
its development.